Open Access

The global existence of a parabolic cross-diffusion system with discontinuous coefficients

Boundary Value Problems20162016:128

https://doi.org/10.1186/s13661-016-0636-1

Received: 21 April 2016

Accepted: 29 June 2016

Published: 11 July 2016

Abstract

This paper deals with a one-dimensional parabolic cross-diffusion system describing two-species models. The system under consideration is strongly coupled and the coefficients of the equations are allowed to be discontinuous. The existence and uniqueness of nonnegative global solutions are given when one of the cross-diffusion pressures is zero. Our approach to the problem is by an approximation method and various estimates.

Keywords

parabolic systemscross-diffusiondiscontinuous coefficientsapproximation method

MSC

35K5535R0535K57

1 Introduction

In the previous work [1], we presented a parabolic cross-diffusion system describing two-species models on a bounded domain with different natural conditions, and studied the corresponding steady-state problem, an elliptic cross-diffusion system. In this paper, we deal with this parabolic cross-diffusion system for the one-dimensional case under the homogeneous Neumann boundary conditions.

Assume that \((0,l)\) is partitioned into two intervals \(I^{(1)}:=(0,l_{0})\) and \(I^{(2)}:=(l_{0},l)\) separated by \(x=l_{0}\), and the natural conditions of \(I^{(i)}\) are different. For any \(T>0\), set
$$\begin{aligned}& Q_{T}:=(0,l)\times(0,T], \\& Q^{(i)}_{T}:=I^{(i)} \times (0,T], \quad i=1,2, \\& S_{T}:=\bigl\{ (x,t):x=0\text{ or } l, t\in[0,T]\bigr\} , \\& \Gamma_{T}:=\bigl\{ (x,t): x=l_{0}, t\in[0,T]\bigr\} . \end{aligned}$$
Let \(u=u(x,t)\), \(v=v(x,t)\) be the population densities of the two species. Assume that the species share the same habitat, the interval \([0,l]\), and that for each species, the density and the flux of the density are all continuous across the inner boundary \(\Gamma_{T}\). According to [1], when the cross-diffusion pressure for the second species is zero, u, v are governed by the parabolic cross-diffusion system
$$ \left \{ \textstyle\begin{array}{l} u_{t}= [k_{1}(x) ((1+\beta u+\gamma v)u )_{x} ]_{x}+uf_{1}(x,u,v)\quad ((x,t)\in Q^{(i)}_{T}), i=1,2, \\ v_{t}= [k_{2}(x) ((1+\delta v)v )_{x} ]_{x}+vf_{2}(x,u,v)\quad ((x,t)\in Q^{(i)}_{T}), i=1,2, \\ u^{-}(l_{0},t)=u^{+}(l_{0},t),\qquad v^{-}(l_{0},t)=v^{+}(l_{0},t)\quad (t\in[0,T]), \\ {[k_{1}(x) ((1+\beta u+\gamma v)u )_{x} ]}^{-}(l_{0},t)= [k_{1}(x) ((1+\beta u+\gamma v)u )_{x} ]^{+}(l_{0},t)\quad (t\in(0,T]), \\ {[k_{2}(x) ((1+\delta v)v )_{x} ]}^{-}(l_{0},t)= [k_{2}(x) ((1+\delta v)v )_{x} ]^{+}(l_{0},t) \quad (t\in(0,T]), \\ u_{x}=v_{x}=0\quad ((x,t)\in S_{T}), \\ u(x,0)=u_{0}(x), \qquad v(x,0)=v_{0}(x)\quad (x\in(0,l)), \end{array}\displaystyle \right . $$
(1.1)
where \(u_{t}:=\partial u/\partial t\), \(u_{x}:=\partial u/\partial x\), \(u^{-}(l_{0},t)\) and \(u^{+}(l_{0},t)\) represent the limits of u from left and right for the space variable, respectively,
$$ f_{j}(x,u,v):=\left \{ \textstyle\begin{array}{l@{\quad}l} f_{j,1}(u,v)& (x\in I^{(1)}), \\ f_{j,2}(u,v)& (x\in I^{(2)}), \end{array}\displaystyle \right .\qquad k_{j}(x):=\left \{ \textstyle\begin{array}{l@{\quad}l} k_{j,1}& (x\in I^{(1)}), \\ k_{j,2}& (x\in I^{(2)}), \end{array}\displaystyle \right . $$
\(k_{j,1}\) and \(k_{j,2}\) (\(j=1,2\)) are positive constants, and β, γ and δ are nonnegative constants. In (1.1), \(k_{1}(x)\) and \(k_{2}(x)\) are the diffusion rates, \(\beta k_{1}(x)\) and \(\delta k_{2}(x)\) are the self-diffusion rates, and \(\gamma k_{1}(x)\) is cross-diffusion rate (see [1]).

In 1979 Shigesada et al. [2] proposed a mathematical model with cross-diffusion to investigate the spatial segregation phenomena of two competing species under inter- and intra-species population pressures. It is a strongly coupled quasilinear parabolic system. Over the past 36 years, quasilinear elliptic and parabolic systems with cross-diffusion have been treated extensively in the literature both in theory and in applications, and most of the treatments are for systems with continuous coefficients (see [316] and the references therein). In [1], by Schauder’s fixed point theorem, we discussed the existence of nonnegative solutions for the elliptic system with cross-diffusion and discontinuous coefficients. In this paper, problem (1.1) is a strongly coupled parabolic system with cross-diffusion. In addition, the diffusion rates, self-diffusion rates, cross-diffusion rates and reaction functions are all allowed to be discontinuous. Based on the result of Lou et al. [6] for a parabolic system with continuous coefficients, we shall use approximation method and various estimates to show the existence and uniqueness of global solutions for problem (1.1).

The paper is organized as follows. In the next section we introduce the notations, hypotheses, and main result. In Section 3 we construct an approximation problem of (1.1) and establish the uniform estimates of solutions for the approximation problem. Section 4 is devoted to the existence and uniqueness of solutions for problem (1.1).

2 The notations, hypotheses, and main result

For any set S, denotes the closure of S. The symbol \(I'\subset\subset I \) means that I and \(I'\) are open intervals and \(\bar{I}'\) is a subset of I. Denote
$$ \|u\|_{L^{s,r}(Q_{T})}:= \biggl[ \int_{0}^{T} \biggl( \int_{0}^{l}\bigl\vert u(x,t)\bigr\vert ^{s}\, \mathrm{d} x \biggr)^{r/s}\, \mathrm{d}t \biggr]^{1/r},\qquad \|u\|_{L^{s}(Q_{T})}:=\|u\|_{L^{s,s}(Q_{T})}, $$
where \(s,r\geq1\). \(W_{2}^{1,0}(Q_{T})\) and \(W_{2}^{1,1}(Q_{T})\) are the Hilbert spaces with scalar products
$$ (u,v)_{W_{2}^{1,0}(Q_{T})}= \int\!\!\! \int_{Q_{T}}(uv+u_{x}v_{x})\, \mathrm {d}x\, \mathrm{d}t,\qquad (u,v)_{W_{2}^{1,1}(Q_{T})}= \int\!\!\! \int _{Q_{T}}(uv+u_{t}v_{t}+u_{x}v_{x}) \, \mathrm{d}x\, \mathrm{d}t, $$
respectively. \(V_{2}(Q_{T})\) is the Banach space consisting of all elements of \(W^{1,0}_{2}(Q_{T})\) having a finite norm
$$ \|u\|_{V_{2}(Q_{T})}:=\sup_{t\in[0,T]}\bigl\Vert u(\cdot,t)\bigr\Vert _{L^{2}(0,l)}+\|u_{x}\| _{L^{2}(Q_{T})}. $$
We let
$$ W^{1}_{r}(0,l):=\bigl\{ u=u(x): u, u_{x}\in L^{r}(0,l) \bigr\} $$
equipped with the norm
$$ \|u\|_{W^{1}_{r}(0,l)}:=\| u\|_{L^{r}(0,l)}+\| u_{x} \|_{L^{r}(0,l)}. $$

Definition 2.1

A nonnegative vector function \((u,v)\) is called a solution of (1.1) if \((u,v)\) possesses the following properties:
  1. (i)

    For any \(T>0\), \(u,v\in W^{1,1}(Q_{T})\cap C^{2,1}(Q^{(i)}_{T})\), \(i=1,2\). For any \(I'\subset\subset (0,l)\), there exists \(\alpha'\in(0,1)\), such that \(u,v\in C^{\alpha'}(\bar{I}'\times[0,T])\). For any fixed \(i\in\{ 1,2\}\) and for any \(I''\subset\subset I^{(i)}\), \(\tau\in(0,T)\), there exists \(\alpha''\in(0,1)\), such that \(u,v\in C^{2+\alpha'',1+\alpha''/2}(\bar{I}''\times[\tau,T])\).

     
  2. (ii)

    For any \(I'\subset\subset (0,l)\) and \(\tau\in(0,T)\), there exists \(\bar{\alpha}'\in(0,1)\), such that \(u_{t},v_{t}\in C^{\bar{\alpha}'}(\bar{I}'\times[\tau,T])\) and \(u_{x},v_{x}\in C^{\bar{\alpha}'}((\bar{I}'\cap\bar{I}^{(i)})\times[\tau,T])\), \(i=1,2\).

     
  3. (iii)

    \((u,v)\) satisfies pointwise the equations in (1.1) on \(Q^{(i)}_{T}\) (\(i=1,2\)), the initial conditions on \(\{(x,t):x\in (0,l), t=0\}\) and the inner boundary conditions on \(\Gamma_{T}\), and satisfies the homogeneous Neumann boundary conditions on \(S_{T}\) for almost all t.

     

We see from property (ii) in Definition 2.1 that if \((u,v)\) is a solution of (1.1), then \(u_{x}\), \(v_{x}\) are Hölder continuous up to the inner boundary \(\Gamma_{T}\), and \(u_{t}\), \(v_{t}\) are Hölder continuous across \(\Gamma_{T}\).

Throughout the paper we make the following hypotheses on the various functions in (1.1):
  1. (H)
    For each \(j,i=1,2\), \(f_{j,i}(u,v)\in C^{1}(\mathbb{R}_{+}^{2} )\), where \(\mathbb{R}_{+}:=[0,+\infty)\). There exist positive constants \(\underline{d}_{1}\), \(\bar{d}_{1}\), \(\underline{e}_{1}\), \(\underline{d}_{2}\), \(\underline{e}_{2}\), \(\bar{e}_{2}\), such that, for any \(u,v\geq0\),
    $$ \left \{ \textstyle\begin{array}{l} -\underline{d}_{1}\leq\frac{\partial f_{1,i}(u,v)}{\partial u}\leq -\bar{d}_{1}, \qquad -\underline{e}_{1}\leq\frac{\partial f_{1,i}}{\partial v}\leq0, \\ -\underline{d}_{2}\leq\frac{\partial f_{2,i}(u,v)}{\partial u}\leq 0,\qquad -\underline{e}_{2}\leq\frac{\partial f_{2,i}}{\partial v}\leq -\bar{e}_{2}. \end{array}\displaystyle \right . $$
    (2.1)
    Assume that \(u_{0}=u_{0}(x)\), \(v_{0}=v_{0}(x)\) possess the properties
    $$ \left \{ \textstyle\begin{array}{l} u_{0}(x),v_{0}(x)\in W^{1}_{r_{0}}(0,l)\cap C^{1}((0,l_{0}])\cap C^{1}([l_{0},l)) \quad \text{for some }r_{0}>2, \\ u_{0}(x),v_{0}(x)\geq0\quad \text{for } x\in[0,l], \end{array}\displaystyle \right . $$
    (2.2)
    and satisfy the compatibility conditions
    $$ \left \{ \textstyle\begin{array}{l} {[k_{1}(x) ((1+\beta u_{0}+\gamma v_{0})u_{0} )_{x} ]}^{-}(l_{0})= [k_{1}(x) ((1+\beta u_{0}+\gamma v_{0})u_{0} )_{x} ]^{+}(l_{0}), \\ {[k_{2}(x) ((1+\delta v_{0})v_{0} )_{x} ]}^{-}(l_{0})= [k_{2}(x) ((1+\delta v_{0})v_{0} )_{x} ]^{+}(l_{0}). \end{array}\displaystyle \right . $$
    (2.3)
     

From (2.2) it follows that \(u_{0}(x),v_{0}(x)\in C^{1/2}([0,l])\).

The main result of this paper is the following theorem.

Theorem 2.1

Let hypothesis (H) hold. Then problem (1.1) has a unique solution \((u,v)\).

Since T is an arbitrary positive number, the solution \((u,v)\) given by Theorem 2.1 is global.

3 An approximation problem of (1.1)

In order to show the existence and uniqueness of solutions for problem (1.1), in this section we shall construct an approximation problem and establish the uniform estimates of its solutions.

3.1 The global solution for the approximation problem

To construct an approximation problem of (1.1), we first construct some approximation functions. For an arbitrary \(\varepsilon>0\), let \(\zeta_{\varepsilon}=\zeta_{\varepsilon}(x)\) a smooth function with values between 0 and 1, such that \(\zeta_{\varepsilon}=1\) for \(x\leq l_{0}\), \(\zeta _{\varepsilon}=0\) for \(x\geq l_{0}+\varepsilon\) and \(|\zeta_{\varepsilon x}|\leq C/\varepsilon\) for all \(x\in \mathbb{R}\). For each \(j=1,2\), define
$$ \left \{ \textstyle\begin{array}{l} k_{j\varepsilon}(x):=\zeta_{\varepsilon}(x)k_{j,1}+(1-\zeta _{\varepsilon}(x))k_{j,2} \quad (x\in [0,l]), \\ f_{j\varepsilon}(x,u,v):=\zeta_{\varepsilon}(x)f_{j,1}(u,v)+(1-\zeta _{\varepsilon}(x))f_{j,2}(u,v)\quad ((x,u,v)\in[0,l]\times \mathbb{R}_{+}^{2}). \end{array}\displaystyle \right . $$
(3.1)
Then (2.1) in hypothesis (H) implies that, for all \((x,u,v)\in[0,l]\times\mathbb{R}_{+}^{2}\),
$$\begin{aligned}& \mu_{0}:=\min_{j,m=1,2}k_{j,m}\leq k_{j\varepsilon }(x)\leq\mu _{1}:=\max_{j,m=1,2}k_{j,m}, \end{aligned}$$
(3.2)
$$\begin{aligned}& \left \{ \textstyle\begin{array}{l} -\underline{d}_{1}\leq\frac{\partial f_{1\varepsilon }(x,u,v)}{\partial u}\leq -\bar{d}_{1},\qquad -\underline{e}_{1}\leq\frac{\partial f_{1\varepsilon }}{\partial v}\leq0, \\ -\underline{d}_{2}\leq\frac{\partial f_{2\varepsilon }(x,u,v)}{\partial u}\leq0,\qquad -\underline{e}_{2}\leq\frac{\partial f_{2\varepsilon}}{\partial v}\leq -\bar{e}_{2}, \end{array}\displaystyle \right . \end{aligned}$$
(3.3)
and
$$ \left \{ \textstyle\begin{array}{l} \min_{i=1,2}f_{1,i}(0,0)-\underline{d}_{1}u -\underline{e}_{1}v\leq f_{1\varepsilon}(x,u,v)\leq\max_{i=1,2}f_{1,i}(0,0)-\bar {d}_{1}u, \\ \min_{i=1,2}f_{2,i}(0,0)-\underline{d}_{2}u -\underline{e}_{2}v\leq f_{2\varepsilon}(x,u,v)\leq \max_{i=1,2}f_{2,i}(0,0)-\bar{e}_{2}v. \end{array}\displaystyle \right . $$
(3.4)
By employing \(k_{j\varepsilon}(x)\), \(f_{j\varepsilon}(x,u,v)\), we consider the following approximation problem of (1.1):
$$ \left \{ \textstyle\begin{array}{l} u_{t}= [k_{1\varepsilon}(x) ((1+\beta u+\gamma v)u )_{x} ]_{x}+uf_{1\varepsilon}(x,u,v)\quad ((x,t)\in Q_{T}), \\ v_{t}= [k_{2\varepsilon}(x) ((1+\delta v)v )_{x} ]_{x}+vf_{2\varepsilon}(x,u,v) \quad ((x,t)\in Q_{T}), \\ u_{x}=v_{x}=0\quad ((x,t)\in S_{T}), \\ u(x,0)=u_{0}(x),\qquad v(x,0)=v_{0}(x) \quad (x\in(0,l)). \end{array}\displaystyle \right . $$
(3.5)

Proposition 3.1

Let hypothesis (H) be satisfied. Then problem (3.5) has a unique nonnegative global solution \((u_{\varepsilon}, v_{\varepsilon})=(u_{\varepsilon}(x,t), u_{\varepsilon}(x,t))\) satisfying \(u_{\varepsilon}(\cdot, t), v_{\varepsilon}(\cdot, t)\in C([0,\infty), W^{1}_{r_{0}}(0,l))\cap C^{\infty}((0,\infty),C^{\infty}(0,l))\).

Proof

By (2.2) and (3.3), and by a minor modification, we can use the arguments in [6] to prove this proposition. The detailed proofs are omitted here. □

3.2 Uniform estimates of \(\max_{Q_{T}}u_{\varepsilon}\), \(\max_{Q_{T}}v_{\varepsilon}\)

In order to investigate the limit of vector sequence \(\{(u_{\varepsilon},v_{\varepsilon})\}\) governed by (3.5), in the rest of this sections we derive some uniform estimates of \(u_{\varepsilon}\), \(v_{\varepsilon}\). We shall use the following relations several times:
$$\begin{aligned}& \bigl((1+\beta_{\varepsilon}u_{\varepsilon}+\gamma v_{\varepsilon})u_{\varepsilon}\bigr)_{x}=(1+2\beta u_{\varepsilon}+\gamma v_{\varepsilon})u_{\varepsilon x}+ \gamma u_{\varepsilon}v_{\varepsilon x}, \\& \bigl((1+\delta v_{\varepsilon})v_{\varepsilon}\bigr)_{x}=(1+2\delta v_{\varepsilon})v_{\varepsilon x}. \end{aligned}$$

For notational simplicity, we shall use \(\alpha'\), \(\alpha''\), , \(C_{0}\), \(C_{1}\) and \(C(\cdots)\) to denote constants depending only on T, β, γ, δ, \(k_{j,m}\), \(f_{j,m}(0,0)\), \(\underline{d}_{j}\), \(\underline{e}_{j}\) (\(j,m=1,2\)), \(\bar{d}_{1}\), \(\bar{e}_{2}\), \(\|u_{0}\|_{W^{1}_{r_{0}}(0,l)}\), \(\|v_{0}\|_{W^{1}_{r_{0}}(0,l)}\) and the quantities appearing in parentheses, independent of ε. The same letter C will be used to denote different constants depending on the same set of arguments.

Lemma 3.2

We have
$$\begin{aligned}& \max_{\bar{Q}_{T}} v_{\varepsilon}(x,t)\leq C_{0}:=\max \Bigl\{ \max_{x\in[0,l]}v_{0},\max_{i=1,2} f_{2,i}(0,0)\big/\bar{e}_{2}\Bigr\} , \end{aligned}$$
(3.6)
$$\begin{aligned}& \max_{t\in[0,T]}\bigl\Vert u_{\varepsilon}(\cdot,t)\bigr\Vert _{L^{1}(0,l)}+\| u_{\varepsilon}\| _{L^{2}(Q_{T})}\leq C, \end{aligned}$$
(3.7)
$$\begin{aligned}& \|v_{\varepsilon}\|_{V_{2}(Q_{T})}\leq C. \end{aligned}$$
(3.8)

Proof

By using the maximum principle (see [17]) and (3.2)-(3.4), the proofs similar to those of Lemmas 2.1, 2.2, 2.4 in [6] imply (3.6)-(3.8). □

Lemma 3.3

The following estimates hold:
$$\begin{aligned}& \sup_{t\in[0,T]}\bigl\Vert v_{\varepsilon x}(\cdot,t)\bigr\Vert _{L^{2}(0,l)}+\Vert v_{\varepsilon t}\Vert _{L^{2}(Q_{T})} \leq C, \end{aligned}$$
(3.9)
$$\begin{aligned}& \Vert v_{\varepsilon x}\Vert _{L^{6}(Q_{T})}+\Vert v_{\varepsilon x}\Vert _{L^{\infty ,4}(Q_{T})}\leq C. \end{aligned}$$
(3.10)

Proof

Let \(w_{\varepsilon}=(1+\delta v_{\varepsilon})v_{\varepsilon}\). Then
$$ w_{\varepsilon t}=(1+2\delta v_{\varepsilon})v_{\varepsilon t},\qquad w_{\varepsilon x}=(1+2\delta v_{\varepsilon})v_{\varepsilon x}, $$
(3.11)
and
$$ w_{\varepsilon t}=(1+2\delta v_{\varepsilon}) \bigl[ \bigl(k_{2\varepsilon }(x)w_{\varepsilon x} \bigr)_{x}+v_{\varepsilon}f_{2\varepsilon }(x,u_{\varepsilon},v_{\varepsilon}) \bigr]\quad \bigl((x,t)\in Q_{T}\bigr). $$
(3.12)
Recall that \(v_{\varepsilon}(\cdot, t)\in C([0,\infty ),W^{1}_{r_{0}}(0,l))\cap C^{\infty}((0,\infty),C^{\infty}(0,l))\). Multiplying (3.12) by \((k_{2\varepsilon}(x)w_{\varepsilon x} )_{x}\), integrating it over \(Q_{t}\) and then using (3.4), (3.6), (3.7), and Cauchy’s inequality, we see that, for any \(\vartheta>0\),
$$\begin{aligned}& \frac{1}{2} \int_{0}^{l} k_{2\varepsilon }(x)w^{2}_{\varepsilon x}(x,t) \,\mathrm{d}x+ \int\!\!\! \int_{Q_{t}}(1+2\delta v_{\varepsilon}) \bigl(k_{2\varepsilon}(x)w_{\varepsilon x} \bigr)^{2}_{x}\,\mathrm{d}x\,\mathrm{d}t \\& \quad =\frac{1}{2} \int_{0}^{l} k_{2\varepsilon}(x)w^{2}_{\varepsilon x}(x,0) \,\mathrm{d}x- \int\!\!\! \int_{Q_{t}}(1+2\delta v_{\varepsilon}) \bigl(k_{2\varepsilon}(x)w_{\varepsilon x} \bigr)_{x}v_{\varepsilon}f_{2\varepsilon}(x,u_{\varepsilon},v_{\varepsilon}) \,\mathrm{d}x\,\mathrm {d}t \\& \quad \leq C \int_{0}^{l} v_{0x}^{2}\, \mathrm{d} x+\vartheta \int\!\!\! \int_{Q_{t}} \bigl(k_{2\varepsilon }(x)w_{\varepsilon x} \bigr)^{2}_{x}\,\mathrm{d}x\,\mathrm{d} t+C(\vartheta) \int\!\!\! \int_{Q_{t}} \bigl(1+u^{2}_{\varepsilon}\bigr)\, \mathrm {d}x\,\mathrm{d}t \\& \quad \leq C(\vartheta)+ \vartheta \int\!\!\! \int_{Q_{t}} \bigl(k_{2\varepsilon }(x)w_{\varepsilon x} \bigr)^{2}_{x}\,\mathrm{d}x\,\mathrm{d}t. \end{aligned}$$
We choose \(\vartheta=1/2\) and employ (3.12) to find
$$ \sup_{t\in[0,T]}\bigl\Vert w_{\varepsilon x}(\cdot,t)\bigr\Vert _{L^{2}(0,l)}+ \bigl\Vert \bigl(k_{2\varepsilon}(x)w_{\varepsilon x} \bigr)_{x}\bigr\Vert _{L^{2}(Q_{T})}+ \Vert w_{\varepsilon t} \Vert _{L^{2}(Q_{T})}\leq C. $$
(3.13)
Furthermore, from (3.13), (3.11), and (3.8) it follows that (3.9) holds, and
$$ \bigl\Vert k_{2\varepsilon}(x)w_{\varepsilon x}\bigr\Vert _{V_{2}(Q_{T})}\leq C, $$
and from Chapter II, equation (3.8) in [18] that
$$ \Vert w_{\varepsilon x}\Vert _{L^{6}(Q_{T})}+ \Vert w_{\varepsilon x} \Vert _{L^{\infty ,4}(Q_{T})}\leq C. $$
Combining this inequality and (3.11) leads us to estimate (3.10). □

Based on Lemma 3.3, let us estimate \(\max_{Q_{T}}u_{\varepsilon}\).

Lemma 3.4

We have
$$ u_{\varepsilon}(x,t)\leq C_{1}\quad \bigl((x,t)\in Q_{T} \bigr). $$
(3.14)

Proof

Choose \(\hat{\sigma}>\max\{\max_{x\in [0,l]}u_{0}(x),1\}\), and let
$$ A_{\sigma}(t):=\bigl\{ x:u_{\varepsilon}>\sigma,x\in[0,l]\bigr\} ,\qquad u^{(\sigma)}_{\varepsilon}:=\max\{u_{\varepsilon}-\sigma;0\}, \quad \sigma \geq\hat{\sigma}. $$
Multiplying the first equation of (3.5) by \(u^{(\sigma)}_{\varepsilon}\) and then integrating by parts over \(Q_{t_{1}}\), we see from (3.2), (3.4), and Cauchy’s inequality that, for any \(\vartheta>0\),
$$\begin{aligned}& \frac{1}{2} \int_{0}^{l} \bigl[u^{(\sigma)}_{\varepsilon}(x,t_{1})\bigr]^{2}\,\mathrm{d}x+ \int_{0}^{t_{1}}\!\! \int_{A_{\sigma}(t) }k_{1\varepsilon}(x) (1+2\beta u_{\varepsilon}+ \gamma v_{\varepsilon})u^{2}_{\varepsilon x}\,\mathrm{d}x\,\mathrm{d} t \\& \quad =- \int_{0}^{t_{1}}\!\! \int_{A_{\sigma}(t) }k_{1\varepsilon }(x)\gamma u_{\varepsilon}u_{\varepsilon x}v_{\varepsilon x}\,\mathrm{d}x\,\mathrm{d}t+ \int _{0}^{t_{1}}\!\! \int_{A_{\sigma}(t) } u_{\varepsilon}u^{(\sigma)}_{\varepsilon}f_{1\varepsilon }(x,u_{\varepsilon},v_{\varepsilon})\,\mathrm{d}x\,\mathrm{d}t \\& \quad \leq C \int_{0}^{t_{1}}\!\! \int_{A_{\sigma}(t) } u_{\varepsilon}|u_{\varepsilon x}||v_{\varepsilon x}| \,\mathrm {d}x\,\mathrm{d}t+ \max_{i=1,2}f_{1,i}(0,0) \int_{0}^{t_{1}}\!\! \int_{A_{\sigma}(t) } u_{\varepsilon}(u_{\varepsilon}-\sigma) \, \mathrm{d}x\,\mathrm{d}t \\& \quad \leq \vartheta \int_{0}^{t_{1}} \int_{A_{\sigma}(t) } u^{2}_{\varepsilon x}\,\mathrm {d}x\, \mathrm{d}t +C(\vartheta) \int_{0}^{t_{1}}\!\! \int_{A_{\sigma}(t) }\bigl(v^{2}_{\varepsilon x}+1\bigr) \bigl[(u_{\varepsilon}-\sigma)^{2}+\sigma^{2}\bigr]\, \mathrm{d}x\,\mathrm{d}t. \end{aligned}$$
Setting \(\vartheta=\mu_{0}/2\), we have
$$ \bigl\Vert u^{(\sigma)}_{\varepsilon}\bigr\Vert ^{2}_{V_{2}(Q_{t_{1}})} \leq C\bigl\Vert \mathcal {D}(x,t)\bigr\Vert _{L^{\bar{s},\bar{r}}(Q_{t_{1}}(\sigma) )} \bigl\Vert (u_{\varepsilon}-\sigma)^{2}+\sigma^{2}\bigr\Vert _{L^{\bar{s}/(\bar{s}-1),\bar {r}/(\bar {r}-1)}(Q_{t_{1}}(\sigma) )} $$
(3.15)
for \(\bar{s}=\bar{r}=2\), where \(\mathcal{D}(x,t):=v^{2}_{\varepsilon x}+1\), \(Q_{t_{1}}(\sigma):=\{(x,t):x\in A_{\sigma}(t),t\in(0,t_{1}] \}\). Thus \(1/\bar{r}+1/(2\bar{s})=1-\chi_{1}\) for \(\chi_{1}=1/4\). In view of (3.10), we find that (3.15) has the same property as [18], Chapter III, equation (7.8). By using Chapter II, Theorem 6.1 and Remark 6.2 in [18], the proof similar to that of Chapter III, equation (7.15) in [18] leads to (3.14). □

3.3 Uniform Hölder estimates

In this subsection, we establish uniform Hölder estimates of \(u_{\varepsilon}\), \(v_{\varepsilon}\) and their derivatives.

Lemma 3.5

For any open interval \(I'\subset\subset (0,l)\), there exists \(\alpha'=\alpha'(d')\in(0,1)\) such that
$$ \|u_{\varepsilon},v_{\varepsilon}\|_{C^{\alpha'}(\bar {I}'\times [0,T])}\leq C \bigl(d'\bigr). $$
(3.16)
Here and below, \(d':=\min\{\operatorname{dist}(0,I'),\operatorname{dist}(l,I')\}\).

Proof

It is obvious that \(u_{\varepsilon}\) is a bounded generalized solution of equation \(u_{t}-(\bar{a}_{\varepsilon}(x,t,u_{x}))_{x}+a_{\varepsilon}(x,t)=0\) and \(v_{\varepsilon}\) is a bounded generalized solution of equation \(v_{t}-(\bar{b}_{\varepsilon}(x,t,v_{x}))_{x}+b_{\varepsilon}(x,t)=0\) in the sense of Chapter V, Section 1 in [18], where
$$\begin{aligned}& \bar{a}_{\varepsilon}(x,t,p)=k_{1\varepsilon}(x)\bigl[1+2\beta u_{\varepsilon}(x,t)+\gamma v_{\varepsilon}(x,t)\bigr]p+k_{1\varepsilon}(x) \gamma v_{\varepsilon x}(x,t)u_{\varepsilon}(x,t), \\& a_{\varepsilon}(x,t)=-u_{\varepsilon}(x,t)f_{1\varepsilon } \bigl(x,u_{\varepsilon}(x,t),v_{\varepsilon}(x,t)\bigr), \\& \bar{b}_{\varepsilon}(x,t,q)=k_{2\varepsilon}(x) \bigl(1+2\delta v_{\varepsilon}(x,t)\bigr)q, \\& b_{\varepsilon}(x,t)=-v_{\varepsilon}(x,t)f_{2\varepsilon } \bigl(x,u_{\varepsilon}(x,t),v_{\varepsilon}(x,t)\bigr). \end{aligned}$$
By (3.2), (3.6), and (3.14), we get
$$\begin{aligned}& \bar{a}_{\varepsilon}(x,t,p)p\geq\frac{\mu_{0}}{2} p^{2}-C \varphi_{0}(x,t), \\& \bigl\vert \bar{a}_{\varepsilon}(x,t,p) \bigr\vert \leq C\vert p\vert +C\varphi_{1}(x,t) \quad \bigl((x,t,p)\in Q_{T}\times\mathbb{ R}\bigr), \\& \bar{b}_{\varepsilon}(x,t,q)q\geq\mu_{0} q^{2},\qquad \bigl\vert \bar{b}_{\varepsilon}(x,t,q)\bigr\vert \leq C\vert q\vert \quad \bigl((x,t,q)\in Q_{T}\times\mathbb{ R}\bigr), \\& \bigl\vert a_{\varepsilon}(x,t)\bigr\vert + \bigl\vert b_{\varepsilon}(x,t)\bigr\vert \leq C \quad \bigl((x,t)\in Q_{T} \bigr), \end{aligned}$$
where \(\varphi_{0}(x,t):= v^{2}_{\varepsilon x}(x,t)\) and \(\varphi _{1}(x,t):= |v_{\varepsilon x}(x,t)|\). Choosing \(s=r=2\), we find from (3.10) that
$$ \|\varphi_{0}\|_{L^{s,r}(Q_{T})}\leq C, \qquad \|\varphi_{1} \| _{L^{2s,2r}(Q_{T})}\leq C. $$
In view of \(u_{0}(x),v_{0}(x)\in C^{1/2}([0,l])\), by employing Chapter V, Section 1, Theorem 1.1 in [18], we obtain (3.16). □

We next give the Hölder estimates of derivatives.

Lemma 3.6

For any fixed \(i\in\{1,2\}\) and for any \(I''\subset\subset I^{(i)}\), \(\tau\in(0,T)\), there exists \(\alpha''=\alpha''(d'',\tau )\in (0,1)\) such that
$$ \|u_{\varepsilon},v_{\varepsilon}\|_{C^{2+\alpha'',1+\alpha ''/2}(\bar{I}''\times [\tau,T])}\leq C \bigl(d'',\tau\bigr). $$
(3.17)
Hereafter, \(d'':=\operatorname{dist}(\partial I^{(i)} ,I'')\).

Proof

Choose open intervals \(I''_{j}\) and positive numbers \(\tau_{j}\), \(j=1,\ldots,4\), such that \(I''\subset\subset I''_{4}\subset\subset \cdots\subset\subset I''_{1}\subset\subset I^{(i)}\) and \(\tau_{1}<\cdots <\tau_{4}<\tau\). By (3.1), for small enough ε, we have
$$ k_{j\varepsilon}(x)=k_{j,i}, \qquad f_{j\varepsilon}(x,u,v)=f_{j,i}(u,v) \quad \bigl(x\in I''_{1}\bigr), j=1,2. $$
We first derive uniform Hölder estimates of derivatives of \(v_{\varepsilon}\). Set
$$ \bar{B}_{\varepsilon}= \bar{B}_{\varepsilon}(x,t,v, q)=k_{2,i}(1+2 \delta v)q,\qquad B_{\varepsilon}= B_{\varepsilon}(x,t,v)=v f_{2,i} \bigl(u_{\varepsilon}(x,t),v\bigr). $$
Then \(v_{\varepsilon}\) is a solution of equation
$$ v_{t}=\bigl(\bar{B}_{\varepsilon}(x,t,v, v_{x}) \bigr)_{x}+B_{\varepsilon}(x,t,v) \quad \bigl((x,t)\in I''_{1}\times(\tau_{1},T]\bigr). $$
Note that
$$ \bar{B}_{\varepsilon q}(x,t,v, q)=k_{2,i}(1+2\delta v),\qquad \bar {B}_{\varepsilon v}=2k_{2,i}\delta q, \qquad \bar{B}_{ \varepsilon x}=0. $$
Thus when \((x,t,v,q)\in I''_{1}\times(\tau_{1},T]\times[0,C_{0}]\times \mathbb{R}\),
$$ \mu_{0}\leq\bar{B}_{\varepsilon q}(x,t,v, q)\leq C,\qquad | \bar{B}_{ \varepsilon v}|\leq C|q|, \qquad \bigl\vert B_{\varepsilon}(x,t,v) \bigr\vert \leq C. $$
According to Chapter V, Theorem 3.1 in [18], it follows that, for some \(\alpha_{1}''=\alpha_{1}''(d'',\tau)\in(0,1)\),
$$ \|v_{\varepsilon x}\|_{C^{\alpha''_{1}}(\bar{I}''_{2}\times[\tau _{2},T])}\leq C\bigl(d'', \tau\bigr). $$
(3.18)
In addition, \(v_{\varepsilon}\) is a solution of the linear equation
$$ v_{t}-\bar{h}_{\varepsilon}(x,t)v_{xx}+ \hat{h}_{\varepsilon}(x,t)v_{x}=h_{\varepsilon}(x,t)\quad \bigl((x,t) \in I''_{2}\times(\tau_{2},T]\bigr), $$
where \(\bar{h}_{\varepsilon}(x,t)=k_{2,i}(1+2\delta v_{\varepsilon}(x,t))\), \(\hat{h}_{\varepsilon}(x,t)=-2k_{2,i}\delta v_{\varepsilon x}\) and \(h_{\varepsilon}(x,t)=v_{\varepsilon}f_{2,i}(u_{\varepsilon},v_{\varepsilon})\). Estimates (3.16), (3.18) imply that, for some \(\alpha''_{2}=\alpha''_{2}(d'',\tau)\in(0,1)\),
$$ \|\hat{h}_{\varepsilon}, \hat{h}_{\varepsilon}, h_{\varepsilon}\|_{C^{\alpha''_{2}}(\bar{I}''_{2}\times[\tau_{2},T])}\leq C\bigl(d'',\tau\bigr). $$
The Schauder estimate for linear parabolic equation further yields
$$ \|v_{\varepsilon }\|_{C^{2+\alpha''_{2},1+\alpha''_{2}/2}(\bar{I}''_{3}\times[\tau _{3},T])}\leq C\bigl(d'', \tau\bigr). $$
(3.19)
We next use (3.19) to show the uniform Hölder estimates of derivatives of \(u_{\varepsilon}\). Let
$$\begin{aligned}& \bar{A}_{\varepsilon}= \bar{A}_{\varepsilon}(x,t,u, p):=k_{1,i} \bigl(1+2\beta u+\gamma v_{\varepsilon}(x,t)\bigr)p+k_{1,i}\gamma u v_{\varepsilon x}(x,t), \\& A_{\varepsilon}=A_{\varepsilon}(x,t,u):=u f_{1,i} \bigl(u,v_{\varepsilon}(x,t)\bigr). \end{aligned}$$
Thus \(u_{\varepsilon}=u_{\varepsilon}(x,t)\) is a solution of the following equation:
$$ u_{ t}= \bigl(\bar{A}_{\varepsilon}(x,t,u, u_{ x}) \bigr)_{x}+A_{\varepsilon}(x,t,u)\quad \bigl((x,t)\in I''_{3} \times(\tau_{3},T]\bigr). $$
Since
$$\begin{aligned}& \bar{A}_{\varepsilon p}=k_{1,i}\bigl(1+2\beta u+\gamma v_{\varepsilon}(x,t)\bigr),\qquad \bar{A}_{\varepsilon u}=2k_{1,i} \beta p+k_{1,i} \gamma v_{\varepsilon x}(x,t), \\& \bar{A}_{\varepsilon x}=k_{1,i}\gamma v_{\varepsilon x}(x,t)p+k_{1,i} \gamma u v_{\varepsilon xx}(x,t), \end{aligned}$$
then we see from (3.6), (3.14), and (3.19) that when \((x,t,u,p)\in I''_{3}\times(\tau_{3},T]\times[0,C_{1}]\times \mathbb{R}\),
$$\begin{aligned}& \mu_{0}\leq\bar{A}_{\varepsilon p}(x,t,u, p)\leq C,\qquad | \bar{A}_{\varepsilon}|+|\bar{A}_{\varepsilon u}|+|\bar {A}_{\varepsilon x}|\leq C \bigl(d'',\tau\bigr) \bigl(\vert p\vert +1\bigr), \\& \bigl\vert A_{\varepsilon}(x,t,u)\bigr\vert \leq C. \end{aligned}$$
Again by Chapter V, Theorem 3.1 in [18] we see that, for some \(\alpha''_{3}=\alpha''_{3}(d'',\tau)\in(0,1)\),
$$ \|u_{\varepsilon x}\|_{C^{\alpha''_{3},\alpha''_{3}/2}(\bar{I}''_{4}\times[\tau _{4},T])}\leq C\bigl(d'', \tau\bigr). $$
(3.20)
Furthermore, \(u_{\varepsilon}\) is a solution of the linear equation
$$ u_{t}-\bar{g}_{\varepsilon}(x,t)u_{xx}+ \hat{g}_{\varepsilon}(x,t)u_{x}+\tilde{g}_{\varepsilon}(x,t)u=g_{\varepsilon}(x,t)\quad \bigl((x,t)\in I''_{4} \times(\tau_{4},T]\bigr), $$
where
$$\begin{aligned}& \bar{g}_{\varepsilon}(x,t):=k_{1,i}\bigl[1+2\beta u_{\varepsilon}(x,t)+\gamma v_{\varepsilon}(x,t)\bigr], \\& \hat{g}_{\varepsilon}(x,t):=-2k_{1,i} \bigl[\beta u_{\varepsilon x}(x,t)+\gamma v_{\varepsilon x}(x,t)\bigr], \\& \tilde{g}_{\varepsilon}(x,t):=-k_{1,i}\gamma v_{\varepsilon xx}(x,t), \\& g_{\varepsilon}(x,t):=u_{\varepsilon}(x,t) f_{1,i} \bigl(u_{\varepsilon}(x,t),v_{\varepsilon}(x,t) \bigr). \end{aligned}$$
Using (3.16), (3.19), and (3.20) leads to
$$ \|\bar{g}_{\varepsilon}, \hat{g}_{\varepsilon},\tilde{g}_{\varepsilon}, g_{\varepsilon}\|_{ C^{\alpha''_{4}}(\bar{I}''_{4}\times[\tau_{4},T])}\leq C\bigl(d'', \tau\bigr) $$
for some \(\alpha''_{4}=\alpha''_{4}(d'',\tau)\in(0,1)\). Again by Schauder estimate we have
$$ \|u_{\varepsilon }\|_{C^{2+\alpha''_{4},1+\alpha''_{4}/2}(\bar{I}''\times[\tau,T])}\leq C\bigl(d'', \tau\bigr), $$
which along with (3.19) yields (3.17). □

3.4 Estimates of \(\|u_{\varepsilon_{1}}-u_{\varepsilon_{2}}, v_{\varepsilon_{1}}-v_{\varepsilon_{2}}\|_{V_{2}(Q_{T})}\)

We first give the following uniform estimates of derivatives of \(u_{\varepsilon}\).

Lemma 3.7

We have
$$\begin{aligned}& \sup_{t\in[0,T]}\bigl\Vert u_{\varepsilon x}(\cdot,t)\bigr\Vert _{L^{2}([0,l])}+\Vert u_{\varepsilon t}\Vert _{L^{2}(Q_{T})} \leq C, \end{aligned}$$
(3.21)
$$\begin{aligned}& \Vert u_{\varepsilon x}\Vert _{L^{6}(Q_{T})}+\Vert u_{\varepsilon x}\Vert _{L^{\infty,4}(Q_{T})} \leq C. \end{aligned}$$
(3.22)

Proof

Let \(z_{\varepsilon}=(1+\beta u_{\varepsilon}+\gamma v_{\varepsilon})u_{\varepsilon}\). Then
$$\begin{aligned}& z_{\varepsilon t}=(1+2\beta u_{\varepsilon}+\gamma v_{\varepsilon})u_{\varepsilon t}+ \gamma u_{\varepsilon}v_{\varepsilon t},\qquad z_{\varepsilon x}=(1+2\beta u_{\varepsilon}+\gamma v_{\varepsilon})u_{\varepsilon x}+\gamma u_{\varepsilon}v_{\varepsilon x}, \end{aligned}$$
(3.23)
$$\begin{aligned}& z_{\varepsilon t}=(1+2\beta u_{\varepsilon}+\gamma v_{\varepsilon}) \bigl[ \bigl(k_{1\varepsilon}(x)z_{\varepsilon x} \bigr)_{x}+u_{\varepsilon}f_{1\varepsilon}(x,u_{\varepsilon},v_{\varepsilon}) \bigr]+\gamma u_{\varepsilon}v_{\varepsilon t}\quad \bigl((x,t)\in Q_{T} \bigr). \end{aligned}$$
(3.24)
Multiplying (3.24) by \((k_{1\varepsilon}(x)z_{\varepsilon x} )_{x}\) and then integrating it over \(Q_{t}\), from (3.4), (3.9), (3.14), and Cauchy’s inequality we see that, for any \(\vartheta>0\),
$$\begin{aligned}& \frac{1}{2} \int_{0}^{l} k_{1\varepsilon }(x)z^{2}_{\varepsilon x}(x,t) \,\mathrm{d}x+ \int\!\!\! \int_{Q_{t}}(1+2\beta u_{\varepsilon}+\gamma v_{\varepsilon}) \bigl(k_{1\varepsilon }(x)z_{\varepsilon x} \bigr)^{2}_{x}\,\mathrm{d}x\,\mathrm{d}t \\& \quad =\frac{1}{2} \int_{0}^{l} k_{1\varepsilon}(x)z^{2}_{\varepsilon x}(x,0) \,\mathrm{d}x- \int\!\!\! \int_{Q_{t}} \bigl[(1+2\beta u_{\varepsilon}+\gamma v_{\varepsilon})u_{\varepsilon}f_{1\varepsilon}(x,u_{\varepsilon},v_{\varepsilon}) +\gamma u_{\varepsilon}v_{\varepsilon t} \bigr] \\& \qquad {}\times\bigl(k_{1\varepsilon }(x)z_{\varepsilon x} \bigr)_{x}\,\mathrm{d} x\, \mathrm{d}t \\& \quad \leq C(\vartheta)+ \vartheta \int\!\!\! \int_{Q_{t}} \bigl(k_{1\varepsilon }(x)z_{\varepsilon x} \bigr)^{2}_{x}\,\mathrm{d}x\,\mathrm{d}t. \end{aligned}$$
Choosing \(\vartheta=1/2\) and using (3.24) and (3.9), we further get
$$ \bigl\Vert k_{1\varepsilon}(x)z_{\varepsilon x}\bigr\Vert _{V_{2}(Q_{T})}+ \Vert z_{\varepsilon t}\Vert _{L^{2}(Q_{T})}\leq C. $$
As in the proof of Lemma 3.3, from (3.23), (3.24), (3.9), (3.10), and Chapter II, equation (3.8) in [18] we conclude that (3.21) and (3.22) hold. □

Lemma 3.8

For any \(\varepsilon_{1}, \varepsilon _{2}>0\), we have
$$ \|u_{\varepsilon_{1}}-u_{\varepsilon_{2}}, v_{\varepsilon _{1}}-v_{\varepsilon_{2}}\| _{V_{2}(Q_{T})}\leq C(\varepsilon_{1}+\varepsilon_{2})^{1/4}. $$
(3.25)

Proof

Let
$$ \bar{u}=u_{\varepsilon_{1}}-u_{\varepsilon_{2}}, \qquad \bar{v}=v_{\varepsilon _{1}}-v_{\varepsilon_{2}}, \qquad \bar{\mathbf{w}}=(\bar{u},\bar{v}). $$
Then it follows from the second equation of (3.5) that
$$\begin{aligned}& \bar{v}_{t}- \bigl\{ k_{2\varepsilon_{1}}(x) (1+2\delta v_{\varepsilon_{1}}) \bar{v}_{x} \bigr\} _{x}- \bigl\{ \bigl[k_{2\varepsilon _{1}}(x) (1+2\delta v_{\varepsilon_{1}})-k_{2\varepsilon_{2}}(x) (1+2\delta v_{\varepsilon_{2}}) \bigr]v_{\varepsilon_{2}x} \bigr\} _{x} \\& \quad =J_{1}:=\bar {v}f_{2\varepsilon_{1}}(x,u_{\varepsilon_{1}},v_{\varepsilon_{1}})+ v_{\varepsilon_{2}} \bigl[f_{2\varepsilon_{1}}(x,u_{\varepsilon _{1}},v_{\varepsilon_{1}}) -f_{2\varepsilon_{2}}(x,u_{\varepsilon_{2}},v_{\varepsilon_{2}}) \bigr]. \end{aligned}$$
Multiplying this equation by and integrating by parts over \(Q_{t}\), from (3.6), (3.14), and Cauchy’s inequality we find that, for any \(\vartheta>0\),
$$\begin{aligned}& \frac{1}{2} \int_{0}^{l} \bar{v}^{2}(x,t)\,\mathrm {d}x+ \int\!\!\! \int _{Q_{t}}k_{2\varepsilon_{1}}(x) (1+2\delta v_{\varepsilon_{1}}) \bar {v}^{2}_{x}\,\mathrm{d}x\,\mathrm{d}t \\& \quad = \int\!\!\! \int_{Q_{t}} \bigl[k_{2\varepsilon_{2}}(x) (1+2\delta v_{\varepsilon_{2}})-k_{2\varepsilon _{1}}(x) (1+2\delta v_{\varepsilon_{1}}) \bigr]v_{\varepsilon_{2}x}\bar{v}_{x}\,\mathrm {d}x\,\mathrm{d} t+ \int\!\!\! \int_{Q_{t}}\bar{v}J_{1}\,\mathrm{d}x\,\mathrm{d}t \\& \quad \leq\vartheta \int\!\!\! \int_{Q_{t}}\bar{v}^{2}_{x}\,\mathrm {d}x\, \mathrm{d}t+C(\vartheta) \int\!\!\! \int _{Q_{t}}\psi^{2} \bigl[\bigl(k_{2\varepsilon_{1}}(x)-k_{2\varepsilon_{2}}(x) \bigr)^{2} +\bar{u}^{2}+\bar{v}^{2} \bigr]\, \mathrm{d}x\,\mathrm{d}t, \end{aligned}$$
where \(\psi:=|u_{\varepsilon_{2}x}|+|u_{\varepsilon _{1}x}|+|v_{\varepsilon_{1}x}|+|v_{\varepsilon_{2}x}|+1\). Taking \(\vartheta=\mu_{0}/2\), we get
$$\begin{aligned}& \int_{0}^{l} \bar{v}^{2}(x,t)\, \mathrm{d}x+ \int\!\!\! \int_{Q_{t}}\bar {v}^{2}_{x}\,\mathrm{d}x\, \mathrm{d} t \\& \quad \leq C \int\!\!\! \int_{Q_{t}}\psi^{2} \bigl[\bigl(k_{2\varepsilon _{1}}(x)-k_{2\varepsilon_{2}}(x) \bigr)^{2} +\bar{u}^{2}+\bar{v}^{2} \bigr]\, \mathrm{d}x\,\mathrm{d}t. \end{aligned}$$
(3.26)
Similarity, we see from the first equation of (3.5) that
$$\begin{aligned} \bar{u}_{t} =& \bigl\{ k_{1\varepsilon_{1}}(x) (1+2\beta u_{\varepsilon _{1}}+\gamma v_{\varepsilon_{1}})\bar{u}_{x} \bigr\} _{x} \\ &{} + \bigl\{ \bigl[k_{1\varepsilon_{1}}(x) (1+2\beta u_{\varepsilon _{1}}+\gamma v_{\varepsilon_{1}})-k_{1\varepsilon_{2}}(x) (1+2\beta u_{\varepsilon _{2}}+\gamma v_{\varepsilon_{2}}) \bigr]u_{\varepsilon_{2}x} \bigr\} _{x} \\ &{}+ \bigl\{ k_{1\varepsilon _{1}}(x)\gamma u_{\varepsilon_{1}} v_{\varepsilon_{1}x}-k_{1\varepsilon_{2}}(x) \gamma u_{\varepsilon_{2}} v_{\varepsilon_{2}x} \bigr\} _{x} \\ &{} +\bar{u}f_{1\varepsilon _{1}}(x,u_{\varepsilon_{1}},v_{\varepsilon_{1}})+ u_{\varepsilon_{2}} \bigl[f_{1\varepsilon_{1}}(x,u_{\varepsilon _{1}},v_{\varepsilon_{1}}) -f_{1\varepsilon_{2}}(x,u_{\varepsilon_{2}},v_{\varepsilon_{2}}) \bigr]. \end{aligned}$$
As we have done in the derivation of (3.26), by a direct computation we have
$$\begin{aligned}& \int_{0}^{l} \bar{u}^{2}(x,t)\, \mathrm{d}x+ \int\!\!\! \int_{Q_{t}}\bar {u}^{2}_{x}\,\mathrm{d}x\, \mathrm{d} t \\& \quad \leq C \int\!\!\! \int_{Q_{t}}\bar{v}^{2}_{x}\,\mathrm{d}x\, \mathrm {d}t+ \int\!\!\! \int_{Q_{t}}\psi^{2} \bigl[\bigl(k_{1\varepsilon _{1}}(x)-k_{1\varepsilon_{2}}(x) \bigr)^{2} +\bar{u}^{2}+\bar{v}^{2} \bigr]\, \mathrm{d}x\,\mathrm{d}t. \end{aligned}$$
(3.27)
Combining (3.26) and (3.27) leads us to the inequality
$$\begin{aligned}& \int_{0}^{l} |\bar{\mathbf{w}}|^{2}(x,t)\, \mathrm {d}x+ \int\!\!\! \int _{Q_{t}}|\bar{\mathbf{w}}_{x}|^{2}\, \mathrm{d}x\,\mathrm{d} t \\& \quad \leq C \int\!\!\! \int_{Q_{t}}\psi^{2}\sum_{j=1}^{2} \bigl(k_{j\varepsilon _{1}}(x)-k_{j\varepsilon_{2}}(x)\bigr)^{2}\,\mathrm{d} x \,\mathrm{d}t+C \int\!\!\! \int_{Q_{t}} \psi^{2}|\bar{\mathbf {w}}|^{2} \,\mathrm{d}x\,\mathrm{d} t \\& \quad \leq C \int_{0}^{t} \int_{l_{0}}^{l_{0}+\varepsilon_{1}+\varepsilon_{2}}\psi^{2}\,\mathrm {d}x\, \mathrm{d} t+Cy(t) \\& \quad \leq C(\varepsilon_{1}+\varepsilon_{2})^{1/2} \|\psi\| ^{2}_{L^{4}(Q_{T})}+Cy(t), \end{aligned}$$
(3.28)
where \(y(t):=\int_{0}^{t} \{ \|\psi(\cdot,t)\|^{2}_{L^{\infty}(0,l)}\int_{0}^{l} |\bar{\mathbf {w}}|^{2}\,\mathrm{d} x \}\,\mathrm{d}t\). In view of (3.10) and (3.22), it follows from (3.28) that
$$ \mathrm{d}y(t)/\mathrm{d}t\leq C\bigl\Vert \psi(\cdot,t)\bigr\Vert ^{2}_{L^{\infty}(0,l)}y(t)+C(\varepsilon _{1}+ \varepsilon_{2})^{1/2}\bigl\Vert \psi(\cdot,t)\bigr\Vert ^{2}_{L^{\infty}(0,l)}. $$
Moreover, Gronwall’s inequality leads to
$$\begin{aligned} y(t) \leq& C(\varepsilon_{1}+\varepsilon_{2})^{1/2} \int _{0}^{t}\bigl\Vert \psi(\cdot ,t)\bigr\Vert ^{2}_{L^{\infty}(0,l)}\,\mathrm{d}t\exp \biggl\{ C \int_{0}^{t}\bigl\Vert \psi (\cdot,t)\bigr\Vert ^{2}_{L^{\infty}(0,l)}\,\mathrm{d} t \biggr\} \\ \leq& C(\varepsilon_{1}+ \varepsilon_{2})^{1/2}, \end{aligned}$$
which along with (3.28) implies (3.25). □

3.5 Uniform estimates of \(\|u_{\varepsilon xt},v_{\varepsilon xt}\|_{L^{2}(I'\times(\tau,T])}\)

To get the regularity of the limit function of sequence \(\{(u_{\varepsilon},v_{\varepsilon})\}\), we need to derive the uniform estimates of \(\|u_{\varepsilon xt},v_{\varepsilon xt}\|_{L^{2}(I'\times(\tau,T])}\).

Lemma 3.9

For any open interval \(I'\subset\subset (0,l)\) and any number \(\tau\in(0,T)\), we have
$$\begin{aligned}& \sup_{[\tau,T]}\bigl\Vert u_{\varepsilon t}(\cdot,t)\bigr\Vert _{L^{2}(I')}+\Vert u_{\varepsilon xt}\Vert _{L^{2}(I'\times(\tau,T])}+\Vert u_{\varepsilon tx}\Vert _{L^{2}(I'\times (\tau,T])}\leq C\bigl(d',\tau\bigr), \end{aligned}$$
(3.29)
$$\begin{aligned}& \sup_{[\tau,T]}\bigl\Vert v_{\varepsilon t}(\cdot,t)\bigr\Vert _{L^{2}(I')}+\Vert v_{\varepsilon xt}\Vert _{L^{2}(I'\times(\tau,T]))}+\Vert v_{\varepsilon tx}\Vert _{L^{2}(I'\times (\tau,T])}\leq C\bigl(d',\tau \bigr). \end{aligned}$$
(3.30)

Proof

Choose open intervals \(I'_{j}\) and positive number \(\tau_{j}\), \(j=1,2,3\), such that \(I'\subset\subset I'_{3}\subset \subset I'_{2}\subset\subset I'_{1}\subset\subset(0,l)\) and \(\tau_{1}<\tau_{2}<\tau_{3}<\tau\). Differentiating the second equation of (3.5) with respect to t and noting that \(v_{\varepsilon xt}=v_{\varepsilon tx}\) we get
$$\begin{aligned} v_{\varepsilon tt} =& \bigl[k_{2\varepsilon}(x) (1+2\delta v_{\varepsilon})v_{\varepsilon tx} \bigr]_{x}+ \bigl[2k_{2\varepsilon}(x)\delta v_{\varepsilon t}v_{\varepsilon x} \bigr]_{x} \\ &{}+ \bigl[v_{\varepsilon}f_{2\varepsilon}(x,u_{\varepsilon},v_{\varepsilon})\bigr]_{t} \quad \bigl((x,t)\in Q_{T}\bigr). \end{aligned}$$
(3.31)
Let \(\lambda=\lambda(x,t)\) be an arbitrary smooth function taking values in \([0,1]\) such that \(\lambda=1\) for \((x,t)\in I'_{2}\times(\tau_{2},T]\), \(\lambda=0\) for \(x\notin I'_{1}\) or \(t\leq\tau_{1}\), and \(|\lambda_{x}|+|\lambda_{t}|\leq C/d'+C/\tau\) for all \((x,t)\). Multiplying equation (3.31) by \(v_{\varepsilon t}\lambda^{2}\) and then integrating by parts over \(I'_{1}\times(\tau_{1},t_{1}]\) for \(t_{1}\in(\tau,T]\), from (3.6), (3.9), (3.14), (3.21), and Cauchy’s inequality we deduce that, for any \(\vartheta>0\),
$$\begin{aligned}& \frac{1}{2} \int_{I'_{1}} v^{2}_{\varepsilon t}(x,t_{1}) \lambda ^{2}(t_{1})\,\mathrm{d}x+ \int_{\tau_{1}}^{t_{1}}\!\! \int _{I'_{1}}k_{2\varepsilon }(x) (1+2\delta v_{\varepsilon})v^{2}_{\varepsilon tx} \lambda^{2}\,\mathrm {d}x\,\mathrm{d}t \\& \quad = \int_{\tau_{1}}^{t_{1}}\!\! \int_{I'_{1}} \bigl\{ -2k_{2\varepsilon }(x)\delta v_{\varepsilon t}v_{\varepsilon x}v_{\varepsilon tx}\lambda^{2} + \bigl[v_{\varepsilon}f_{2\varepsilon }(x,u_{\varepsilon},v_{\varepsilon}) \bigr]_{t}v_{\varepsilon t}\lambda^{2} \\& \qquad {}+v^{2}_{\varepsilon t}\lambda\lambda _{t}-2k_{2\varepsilon}(x) \bigl[(1+2\delta v_{\varepsilon})v_{\varepsilon tx}v_{\varepsilon t}+2\delta v^{2}_{\varepsilon t}v_{\varepsilon x} \bigr]\lambda \lambda_{x} \bigr\} \,\mathrm{d}x\,\mathrm{d}t \\& \quad \leq\vartheta \int_{\tau_{1}}^{t_{1}}\!\! \int _{I'_{1}}v^{2}_{\varepsilon tx}\lambda ^{2}\,\mathrm{d}x\,\mathrm{d} t+C(\vartheta) \int_{\tau_{1}}^{t_{1}}\!\! \int_{I'_{1}}v^{2}_{\varepsilon x}v^{2}_{\varepsilon t} \lambda^{2}\,\mathrm{d}x\,\mathrm{d}t+ C\bigl(d',\tau \bigr). \end{aligned}$$
Choosing \(\vartheta=\mu_{0}/4\), we have
$$\begin{aligned}& \int_{I'_{1}} v^{2}_{\varepsilon t}(x,t_{1}) \lambda ^{2}(t_{1})\,\mathrm{d}x+ \int _{\tau_{1}}^{t_{1}}\!\! \int_{I'_{1}}v^{2}_{\varepsilon tx}\lambda^{2} \,\mathrm {d}x\,\mathrm{d} t \\& \quad \leq C\bigl(d',\tau\bigr)+C \int_{\tau_{1}}^{t_{1}} \biggl\{ \bigl\Vert v_{\varepsilon x}(\cdot,t)\bigr\Vert ^{2}_{L^{\infty}(0,l)} \int_{I'_{1}} v^{2}_{\varepsilon t}\lambda^{2} \,\mathrm{d} x \biggr\} \,\mathrm{d}t. \end{aligned}$$
As in the proof of Lemma 3.8, employing Gronwall’s inequality and (3.10), from this inequality we get (3.30) and the uniform estimate of \(\|v_{\varepsilon t}\|_{ V_{2}(I'_{2}\times(\tau_{2},T])}\). Chapter II, equation (3.8) in [18] further implies that
$$ \|v_{\varepsilon t}\|_{L^{6}(I'_{2}\times(\tau _{2},T])}\leq C\bigl(d',\tau\bigr). $$
(3.32)
Based on (3.30) and (3.32), we next show that (3.29) holds. Let \(\xi=\xi(x,t)\) be a smooth function with values between 0 and 1 such that \(\xi=1\) for \((x,t)\in I'_{3}\times(\tau_{3},T]\), \(\xi=0\) for \(x\notin I'_{2}\) or \(t\leq \tau_{2}\), and \(|\xi_{x}|+|\xi_{t}|\leq C/d'+C/\tau\) for all \((x,t)\). Similarly, differentiating the first equation of (3.5) with respect to t, multiplying the resulting equation by \(u_{\varepsilon t}\xi^{2}\) and integrating it over \(I'_{2}\times(\tau_{2},T]\), we find from (3.6), (3.9), (3.14), (3.21), and Cauchy’s inequality that, for any \(\vartheta>0\),
$$\begin{aligned}& \frac{1}{2} \int_{I'_{2}} u^{2}_{\varepsilon t}(x,t_{1}) \xi ^{2}(t_{1})\,\mathrm{d}x+ \int_{\tau_{2}}^{t_{1}}\!\! \int _{I'_{2}}k_{1\varepsilon}(x) (1+2\beta u_{\varepsilon}+ \gamma v_{\varepsilon})u^{2}_{\varepsilon tx}\xi^{2}\,\mathrm {d}x\,\mathrm{d}t \\& \quad = \int_{\tau_{2}}^{t_{1}}\!\! \int_{I'_{2}} \bigl\{ \bigl[u_{\varepsilon}f_{1\varepsilon }(x,u_{\varepsilon},v_{\varepsilon}) \bigr]_{t}u_{\varepsilon t}\xi^{2}+u^{2}_{\varepsilon t} \xi\xi_{t} \\& \qquad {}-k_{1\varepsilon}(x) \bigl[(2\beta u_{\varepsilon t}+\gamma v_{\varepsilon t})u_{\varepsilon x} +\gamma ( u_{\varepsilon t}v_{\varepsilon x}+ u_{\varepsilon }v_{\varepsilon tx} ) \bigr]u_{\varepsilon tx}\xi^{2} \\& \qquad {}-2k_{1\varepsilon}(x) \bigl[(2\beta u_{\varepsilon t}+\gamma v_{\varepsilon t})u_{\varepsilon x} +\gamma ( u_{\varepsilon t}v_{\varepsilon x}+ u_{\varepsilon}v_{\varepsilon tx} ) \bigr]u_{\varepsilon t}\xi\xi_{x} \bigr\} \,\mathrm{d}x\,\mathrm{d}t \\& \quad \leq\vartheta \int_{\tau_{2}}^{t_{1}}\!\! \int _{I'_{2}}u^{2}_{\varepsilon tx}\xi^{2}\, \mathrm{d}x\,\mathrm{d}t+C(\vartheta) \int_{\tau_{2}}^{t_{1}}\!\! \int_{I'_{2}} v^{2}_{\varepsilon tx}\,\mathrm{d}x\, \mathrm{d}t+ C\bigl(d',\tau,\vartheta\bigr) \\& \qquad {}+C\bigl(d',\tau,\vartheta\bigr) \int_{\tau _{2}}^{t_{1}}\!\! \int_{I'_{2}} \bigl(u^{4}_{\varepsilon x}+v^{4}_{\varepsilon t} \bigr)\,\mathrm{d}x\,\mathrm{d} t \\& \qquad {}+C \int_{\tau_{2}}^{t_{1}}\!\! \int_{I'_{2}}u^{2}_{\varepsilon t}\bigl(\vert u_{\varepsilon x}\vert +|v_{\varepsilon x}|\bigr)^{2}\xi^{2} \,\mathrm{d}x\,\mathrm{d}t. \end{aligned}$$
Setting \(\vartheta=\mu_{0}/4\) and using (3.10), (3.22), (3.30), and (3.32), we have
$$\begin{aligned}& \int_{I'_{2}} u^{2}_{\varepsilon t}(x,t_{1}) \xi ^{2}(t_{1})\,\mathrm{d} x+ \int_{\tau_{2}}^{t_{1}}\!\! \int_{I'_{2}}u^{2}_{\varepsilon tx}\xi ^{2}\, \mathrm{d}x\,\mathrm{d} t \\& \quad \leq C\bigl(d',\tau\bigr)+C \int_{\tau_{2}}^{t_{1}} \biggl\{ \bigl\Vert \bigl\vert u_{\varepsilon x}(\cdot,t)\bigr\vert +\bigl\vert v_{\varepsilon x}(\cdot,t) \bigr\vert \bigr\Vert ^{2}_{L^{\infty}(0,l)} \int_{I'_{2}} v^{2}_{\varepsilon t}\xi ^{2}\, \mathrm{d} x \biggr\} \,\mathrm{d}t. \end{aligned}$$
Again by Gronwall’s inequality we obtain (3.29). □

4 The proof of Theorem 2.1

To prove Theorem 2.1, let us discuss the behavior of vector sequence \(\{(u_{\varepsilon},v_{\varepsilon})\}\) governed by (3.5) as \(\varepsilon\to0\).

Lemma 4.1

The sequences \(\{u_{\varepsilon}\}\) and \(\{ v_{\varepsilon}\}\) converge to nonnegative bounded functions u and v in \(V_{2}(Q_{T})\), respectively. Moreover, \((u, v)\) possesses property (i) in Definition  2.1, and satisfies pointwise the equations in (1.1) on \(Q^{(i)}_{T}\) (\(i=1,2\)) and the initial conditions on \(\{(x,t): x\in (0,l), t=0\}\). For any \(\eta\in W^{1,0}_{2}(Q_{T})\), the following integral identities hold:
$$\begin{aligned}& \int\!\!\! \int_{Q_{t}} \bigl[u_{ t}\eta +k_{1}(x) \bigl((1+\beta u+\gamma v)u \bigr)_{x}\eta_{x} \bigr]\, \mathrm{d}x\,\mathrm{d}t= \int\!\!\! \int_{Q_{t}} uf_{1}(x,u,v)\eta\,\mathrm{d} x\, \mathrm{d}t, \end{aligned}$$
(4.1)
$$\begin{aligned}& \int\!\!\! \int_{Q_{t}} \bigl[v_{ t}\eta+k_{2}(x) \bigl((1+\delta v)v \bigr)_{x}\eta_{x} \bigr]\,\mathrm {d}x \,\mathrm{d} t= \int\!\!\! \int_{Q_{t}} v f_{2}(x,u,v)\eta\,\mathrm{d}x\,\mathrm {d}t. \end{aligned}$$
(4.2)

Proof

We see from estimate (3.25) that there exist functions \(u, v\in V_{2}(Q_{T})\) such that
$$ u_{\varepsilon}\to u,\qquad v_{\varepsilon}\to v\quad \text{in } V_{2}(Q_{T}), $$
(4.3)
and from (3.9), (3.21) that there exist subsequences (hereafter we retain the same notations for them) \(\{u_{\varepsilon}\}\), \(\{v_{\varepsilon}\}\), such that
$$ u_{\varepsilon t}\rightharpoonup u_{t},\qquad v_{\varepsilon t} \rightharpoonup v_{t}\quad \text{weakly in } L^{2}(Q_{T}). $$
(4.4)
In addition, it follows from (3.6), (3.14), (3.16), and the Arzela-Ascoli theorem that, for any given open interval \(I'\subset\subset(0,l)\), there exist subsequences \(\{u_{\varepsilon}\}\), \(\{v_{\varepsilon}\}\), such that \(\{u_{\varepsilon}\}\), \(\{v_{\varepsilon}\}\) converge to u, v in \(C(\bar{I}'\times[0,T] )\), respectively, and from (3.17) that, for any given \(i\in\{1,2\}\), \(I''\subset\subset Q^{(i)}_{T}\) and \(\tau\in(0,T)\), there exist subsequences \(\{u_{\varepsilon}\}\), \(\{v_{\varepsilon}\}\), such that \(\{u_{\varepsilon}\}\), \(\{v_{\varepsilon}\}\) converge to u, v in \(C^{2,1}(\bar{I}''\times[\tau,T] )\), respectively. Then u, v are in \(W^{1,1}_{2}(Q_{T})\cap C^{\alpha'}(\bar{I}'\times[0,T])\cap C^{2,1}(Q^{(i)}_{T})\cap C^{2+\alpha'',1+\alpha''/2}(\bar{I}''\times [\tau,T])\). Furthermore, (3.5) implies that \((u,v)\) satisfies pointwise the equations in (1.1) on \(Q^{(i)}_{T}\) (\(i=1,2\)) and the initial conditions on \(\{(x,t):x\in (0,l), t=0\}\). By (3.6), (3.9), (3.14), and (3.21) we have
$$\begin{aligned}& (0,0)\leq(u,v)\leq(C_{1}, C_{0})\quad \bigl((x,t)\in Q_{T}\bigr), \end{aligned}$$
(4.5)
$$\begin{aligned}& \sup_{t\in[0,T]}\bigl\Vert u_{x}(\cdot,t),v_{x}( \cdot,t)\bigr\Vert _{L^{2}(0,l)},\qquad \Vert u_{t},v_{t} \Vert _{L^{2}(Q_{T})} \leq C. \end{aligned}$$
(4.6)
For any \(\eta\in W^{1,0}_{2}(Q_{T})\), multiply the equations in (3.5) by η and then integrate by parts over \(Q_{t}\) to get
$$\begin{aligned}& \int\!\!\! \int_{Q_{t}} \bigl[u_{\varepsilon t}\eta +k_{1\varepsilon}(x) \bigl((1+\beta u_{\varepsilon}+\gamma v)u_{\varepsilon}\bigr)_{x} \eta_{x} \bigr]\,\mathrm{d}x\,\mathrm{d}t \\& \quad = \int\!\! \! \int_{Q_{t}}u_{\varepsilon}f_{1\varepsilon}(x,u_{\varepsilon},v_{\varepsilon}) \eta\,\mathrm {d}x\,\mathrm{d}t, \end{aligned}$$
(4.7)
$$\begin{aligned}& \int\!\!\! \int_{Q_{t}} \bigl[v_{\varepsilon t}\eta+k_{2\varepsilon}(x) \bigl((1+\delta v_{\varepsilon})v_{\varepsilon}\bigr)_{x} \eta_{x} \bigr]\,\mathrm{d}x\,\mathrm {d}t= \int\!\!\! \int_{Q_{t}}v_{\varepsilon}f_{2\varepsilon}(x,u_{\varepsilon},v_{\varepsilon}) \eta\,\mathrm {d}x\,\mathrm{d}t. \end{aligned}$$
(4.8)
We next investigate the limit of each term in (4.7), (4.8). It follows from (3.1), (4.3), and (4.5) that
$$\begin{aligned}& \bigl\Vert u_{\varepsilon}f_{1\varepsilon}(x,u_{\varepsilon},v_{\varepsilon})-u f_{1}(x,u,v)\bigr\Vert ^{2}_{L^{2}(Q_{T})} \\& \quad \leq C\Vert u_{\varepsilon}-u \Vert ^{2}_{L^{2}(Q_{T})}+C \bigl\Vert f_{1\varepsilon }(x,u_{\varepsilon},v_{\varepsilon})-f_{1}(x,u,v)\bigr\Vert ^{2}_{L^{2}(Q_{T})} \\& \quad \leq C \bigl[\Vert u_{\varepsilon}-u \Vert ^{2}_{L^{2}(Q_{T})} +\bigl\Vert f_{1,1}(u_{\varepsilon},v_{\varepsilon})-f_{1,1}(u,v) \bigr\Vert ^{2}_{L^{2}(Q^{(1)}_{T})} \\& \qquad {}+\bigl\Vert f_{1,2}(u_{\varepsilon},v_{\varepsilon})-f_{1,2}(u,v) \bigr\Vert ^{2}_{L^{2}((l_{0}+\varepsilon,l)\times (0,T])} \\& \qquad {}+\bigl\Vert f_{1\varepsilon}(x,u_{\varepsilon},v_{\varepsilon})-f_{1}(x,u,v) \bigr\Vert ^{2}_{L^{2}((l_{0},l_{0}+\varepsilon )\times (0,T])} \bigr] \\& \quad \leq C\bigl(\Vert u_{\varepsilon}-u\Vert ^{2}_{L^{2}(Q_{T})}+ \Vert v_{\varepsilon}-v\Vert ^{2}_{L^{2}(Q_{T})}\bigr)+C \varepsilon \\& \quad \to0 \end{aligned}$$
(4.9)
and
$$\begin{aligned}& \sup_{t\in[0,T]} \bigl\Vert k_{1\varepsilon}(x) (1+2\beta u_{\varepsilon}+\gamma v_{\varepsilon})-k_{1}(x) (1+2\beta u+ \gamma v )\bigr\Vert ^{2}_{L^{2}(0,l)} \\& \quad \leq C\bigl\Vert k_{1\varepsilon}(x)-k_{1}(x)\bigr\Vert ^{2}_{L^{2}(l_{0},l_{0}+\varepsilon )} \\& \qquad {}+C\sup_{t\in[0,T]} \bigl[\bigl\Vert u_{\varepsilon}( \cdot,t)-u(\cdot,t)\bigr\Vert ^{2}_{L^{2}(0,l)}+\bigl\Vert v_{\varepsilon}(\cdot,t)-v(\cdot,t)\bigr\Vert ^{2}_{L^{2}(0,l)} \bigr] \\& \quad \to 0. \end{aligned}$$
(4.10)
Furthermore, (3.22) and (4.10) yield
$$\begin{aligned}& \int\!\!\! \int_{Q_{T}}\bigl\vert k_{1\varepsilon}(x) (1+2\beta u_{\varepsilon}+\gamma v_{\varepsilon})-k_{1}(x) (1+2\beta u+ \gamma v )\bigr\vert ^{2}u^{2}_{\varepsilon x}\,\mathrm{d}x\, \mathrm{d}t \\& \quad \leq\sup_{t\in [0,T]} \int_{0}^{l} \bigl\vert k_{1\varepsilon}(x) (1+2 \beta u_{\varepsilon}+\gamma v_{\varepsilon})-k_{1}(x) (1+2\beta u+\gamma v )\bigr\vert ^{2}\,\mathrm{d}x \\& \qquad {}\times \int_{0}^{T} \bigl\Vert u_{\varepsilon x}(\cdot,t) \bigr\Vert ^{2}_{L^{\infty}([0,l])}\,\mathrm{d}t \\& \quad \to0. \end{aligned}$$
Consequently,
$$\begin{aligned}& \bigl\Vert k_{1\varepsilon}(1+2\beta u_{\varepsilon}+\gamma v_{\varepsilon})u_{\varepsilon x}- k_{1}(1+2\beta u+\gamma v )u_{x}\bigr\Vert ^{2}_{L^{2}(Q_{T})} \\& \quad \leq 2 \bigl\Vert \bigl[k_{1\varepsilon}(1+2\beta u_{\varepsilon}+ \gamma v_{\varepsilon})-k_{1}(1+2\beta u+\gamma v ) \bigr]u_{\varepsilon x}\bigr\Vert ^{2}_{L^{2}(Q_{T})}+C\Vert u_{\varepsilon x}-u_{ x}\Vert ^{2}_{L^{2}(Q_{T})} \\& \quad \to0. \end{aligned}$$
(4.11)
Similar arguments show that
$$\begin{aligned}& v_{\varepsilon}f_{2\varepsilon}(x,u_{\varepsilon},v_{\varepsilon})\to v f_{2}(x,u,v),\qquad \gamma u_{\varepsilon}v_{\varepsilon x}\to\gamma u v_{ x}\quad \text{in } L^{2}(Q_{T}), \end{aligned}$$
(4.12)
$$\begin{aligned}& k_{2\varepsilon}(x) (1+2\delta v_{\varepsilon})v_{\varepsilon x}\to k_{2}(x) (1+2\delta v )v_{ x}\quad \text{in } L^{2}(Q_{T}). \end{aligned}$$
(4.13)

Letting \(\varepsilon\to0\) and using (4.4) and (4.9)-(4.13), we find from (4.7), (4.8) that the integral identities (4.1), (4.2) hold. □

We next investigate the regularity of \((u,v)\) in the neighborhood of inner boundary \(\Gamma_{T}\).

Lemma 4.2

For any \(I'\subset\subset (0,l)\) and \(\tau\in(0,T)\), \(v_{t}\in C^{\bar{\alpha}'}(\bar{I}'\times[\tau ,T])\) and \(v_{x}\in C^{\bar{\alpha}'}((\bar{I}'\cap\bar{I}^{(i)})\times[\tau,T])\) (\(i=1,2\)) for some \(\bar{\alpha}'=\bar{\alpha}'(d',\tau)\in(0,1)\).

Proof

It follows from (3.29), (3.30), (4.3), and (4.6) that there exist subsequences \(\{u_{\varepsilon}\}\), \(\{v_{\varepsilon}\}\), such that
$$ u_{\varepsilon xt}\rightharpoonup u_{xt},\qquad v_{\varepsilon xt} \rightharpoonup v_{xt} \quad \text{weakly in } L^{2}\bigl(I' \times(\tau,T]\bigr), $$
and
$$ \sup_{[\tau,T]}\bigl\Vert u_{t}( \cdot,t),v_{t}(\cdot,t)\bigr\Vert _{L^{2}(I')}, \Vert u_{xt},v_{xt}\Vert _{L^{2}(I'\times(\tau,T])} \leq C \bigl(d',\tau\bigr). $$
(4.14)
Let \(\tilde{B}:= \tilde{B}(x,t,w, q)=k_{2}(x)(1+2\delta w)q\), \(\hat{B}:= \hat{B}(x,t,w)=-wf_{2}(x,u(x,t),w)\). Then \(v(x,t)\) is a generalized solution of the single equation \(w_{t}- (\tilde{B}(x,t,w, w_{x}) )_{x}+\hat{B}(x,t,w)=0\) in the sense of Definition 1.1 in [19]. We find that when \((x,t,w,q)\in(I'\cap I^{(i)})\times(\tau,T]\times[0,C_{0}]\times\mathbb{R}\),
$$\begin{aligned}& \frac{\partial\tilde{B}}{\partial q}=k_{2,i}(1+2\delta w),\qquad \frac{\partial\tilde{B}}{\partial w}=2k_{2,i} \delta q,\qquad \frac{\partial\tilde{B}}{\partial x}= \frac{\partial\tilde {B}}{\partial t}=0, \\& \frac{\partial\hat{B}}{\partial w}=-f_{2,i}\bigl(u(x,t),w\bigr)-w\frac {\partial f_{2,i }(u(x,t),w)}{\partial w}, \\& \hat{B}_{x}=-w\frac{\partial f_{2,i }(u,w)}{\partial u} \bigg|_{u=u(x,t)}u_{x}(x,t), \qquad \hat{B}_{t}=-w\frac{\partial f_{2,i }(u,w)}{\partial u} \bigg|_{u=u(x,t)}u_{t}(x,t). \end{aligned}$$
Hence when \((x,t,w,q)\in(I'\cap I^{(i)})\times(\tau,T]\times [0,C_{0}]\times\mathbb{R}\),
$$\begin{aligned}& \bigl\vert \tilde{B}(x,t,w, q)\bigr\vert \leq C\bigl(1+\vert q\vert \bigr), \qquad \mu_{0}\leq\tilde{B}_{q}\leq C, \qquad \vert \tilde{B}_{w}\vert \leq C\vert q\vert , \\& \bigl\vert \hat{B}(x,t,w)\bigr\vert +\vert \hat{B}_{w}\vert \leq C, \qquad \vert \hat{B}_{x}\vert \leq C\psi_{1}(x,t), \qquad \vert \hat{B}_{t}\vert \leq C\psi_{1}(x,t), \end{aligned}$$
where \(\psi_{1}(x,t):=|u_{x}(x,t)|+|u_{t}(x,t)|\). By (4.6) and (4.14), we have
$$ \sup_{t\in[\tau,T]}\bigl\Vert \psi_{1}(\cdot,t)\bigr\Vert _{L^{2}(0,l)}\leq C\bigl(d',\tau\bigr). $$
According to Theorem 1.1 in [19], there exists \(\bar{\alpha}'_{1}=\bar{\alpha}'_{1}(d',\tau)\in(0,1)\) such that
$$ \|v_{t}\|_{ C^{\bar{\alpha}'_{1}}(\bar{I}'\times[\tau ,T])}\leq C\bigl(d',\tau \bigr). $$
(4.15)
Note that \(v\in C^{\alpha'}(\bar{I}'\times[0,T])\) and \((k_{2,i}(1+2\delta v)v_{x})_{x}=v_{t}-vf_{2,i}(u,v)\) on \(Q^{(i)}_{T}\). We see that \((k_{2,i}(1+2\delta v)v_{x})_{x}\in C^{\bar{\alpha}'_{2}}((\bar{I}'\cap \bar{I}^{(i)})\times[\tau,T])\), and \(k_{2,i}(1+2\delta v)v_{x}\in C^{1+\bar{\alpha}'_{2},\bar{\alpha}'_{2} }((\bar{I}'\cap \bar{I}^{(i)})\times[\tau,T])\) for some \(\bar{\alpha}'_{2}\in(0,1)\). Thus \(v_{x}\in C^{\bar{\alpha}'_{3} }((\bar{I}'\cap \bar{I}^{(i)})\times[\tau,T])\). Since \((k_{2,i}(1+2\delta v)v_{x})_{x}=k_{2,i}(1+2\delta v)v_{xx}+2k_{2,i}\delta v^{2}_{x}\) on \(Q^{(i)}_{T}\),
$$ \|v_{x}, v_{xx}\|_{ C^{\bar{\alpha}'_{3} }(\bar{I}'\cap \bar{I}^{(i)}\times[\tau,T])}\leq C \bigl(d',\tau\bigr). $$
(4.16)
This completes the proof of Lemma 4.2. □

Lemma 4.3

For any \(I'\subset\subset (0,l)\) and \(\tau\in(0,T)\), \(u_{t}\in C^{\tilde{\alpha}'}(\bar{I}'\times[\tau,T])\) and \(u_{x}\in C^{\tilde{\alpha}'}((\bar{I}'\cap \bar{I}^{(i)})\times[\tau,T])\) (\(i=1,2\)) for some \(\tilde{\alpha}'=\tilde{\alpha}'(d',\tau)\in(0,1)\). Furthermore, \((u,v)\) satisfies pointwise the inner boundary conditions in (1.1) on \(\Gamma_{T}\), and satisfies the homogeneous Neumann boundary conditions on \(S_{T}\) for almost all \(t\in[0,T]\).

Proof

To investigate the regularity of u, we need to estimate \(\sup_{t\in[\tau,T]}\|v_{xt}(\cdot,t)\|_{L^{2}(I')}\). Choose open intervals \(I'_{1}\), \(I'_{2}\), \(I'_{3}\) and positive numbers \(\tau_{1}\), \(\tau_{2}\), \(\tau_{3}\), such that \(I'\subset\subset I'_{3}\subset \subset I'_{2}\subset\subset I'_{1}\subset\subset(0,l)\) and \(\tau_{1}<\tau_{2}<\tau_{3}<\tau\). According to Lemma 4.1, \((u,v)\) satisfies pointwise the equations in (1.1) on \(Q^{(i)}_{T}\) (\(i=1,2\)). Furthermore, from (4.2) in Lemma 4.1 and the regularity of v in Lemma 4.2 we conclude that v satisfies pointwise the inner boundary condition \([k_{2}(x) ((1+\delta v)v )_{x} ]^{-}(l_{0},t)= [k_{2}(x) ((1+\delta v)v )_{x} ]^{+}(l_{0},t) \) for \(t\in(0,T]\).

Let \(w=(1+\delta v)v\), \(w_{(t)}:=(w(x,t+\Delta t)-w(x,t))/\Delta t\). Then
$$ \left \{ \textstyle\begin{array}{l} w_{t}=(1+2\delta v)v_{t}=(1+2\delta v) (k_{2}(x)w_{x} )_{x}+(1+2\delta v)vf_{2}(x,u,v), \\ w_{x}=(1+2\delta v)v_{ x},\qquad w_{xt}=(1+2\delta v)v_{xt}+2\delta v_{x}v_{t}, \end{array}\displaystyle \right . $$
(4.17)
and
$$ \left \{ \textstyle\begin{array}{l} -w_{t(t)}+ \{(1+2\delta v) (k_{2}(x)w_{x} )_{x} \}_{(t)}=- \{(1+2\delta v)vf_{2}(x,u,v) \}_{(t)} \\ \quad ((x,t)\in Q^{(i)}_{T}), i=1,2, \\ w^{-}_{(t)}(l_{0},t)=w^{+}_{(t)}(l_{0},t) \quad (t\in (0,T]), \\ {[ (k_{2}(x)w_{x} )_{(t)} ]}^{-}(l_{0},t)= [ (k_{2}(x)w_{x} )_{(t)} ]^{+}(l_{0},t)\quad (t\in(0,T]). \end{array}\displaystyle \right . $$
(4.18)
Let \(\xi=\xi(x,t)\) be a smooth function taking values between 0 and 1 such that \(\xi(x,t)=1\) for \((x,t)\in I'_{2}\times(\tau_{2},T]\), \(\xi=0\) for \(x\notin I'_{1}\) or \(t\leq\tau_{1}\), and \(|\xi_{x}|+|\xi_{t}|\leq C(d',\tau)\) for all \((x,t)\). Since \(v_{\varepsilon xt}=v_{\varepsilon tx}\), then \(v_{ xt}=v_{ tx}\) and \(w_{ xt}=w_{ tx}\) on \(I'_{1}\times(\tau_{1},T] \). Multiplying the equations in (4.18) by \((k_{2}(x)w_{x(t)} )_{x}\xi^{2}\) and integrating by parts over \(I'_{1}\times(\tau_{1},t)\), by direct computation we have
$$\begin{aligned}& \frac{1}{2} \int_{I'_{1}} k_{2}(x)w^{2}_{x(t)} \xi ^{2}\,\mathrm{d} x+ \int_{\tau_{1}}^{t}\!\! \int_{I'_{1}} \bigl[- k_{2}(x)w^{2}_{x(t)} \xi\xi_{t}- 2k_{2}(x)w_{t(t)}w_{x(t)}\xi \xi_{x} \bigr]\,\mathrm{d}x\,\mathrm{d}t \\& \qquad {}+ \int_{\tau_{1}}^{t}\!\! \int_{I'_{1}} \bigl\{ \bigl(1+2\delta v(x,t+\Delta t)\bigr) \bigl(k_{2}(x)w_{x(t)} \bigr)^{2}_{x} \\& \qquad {}+ 2 \delta v_{(t)} \bigl(k_{2}(x)w_{x} \bigr)_{x} \bigl(k_{2}(x)w_{x(t)} \bigr)_{x} \bigr\} \xi^{2}\,\mathrm{d} x\,\mathrm{d}t \\& \quad =- \int_{\tau_{1}}^{t}\!\! \int_{I'_{1}} \bigl[(1+2\delta v)vf_{2}(x,u,v) \bigr]_{(t)} \bigl(k_{2}(x)w_{x(t)} \bigr)_{x}\xi^{2}\,\mathrm {d}x\,\mathrm{d}t. \end{aligned}$$
Employing Cauchy’s inequality we see that, for any \(\vartheta_{1},\vartheta_{2}>0\),
$$\begin{aligned}& \frac{1}{2} \int_{I'_{1}} k_{2}(x)w^{2}_{x(t)} \xi ^{2}\,\mathrm{d} x + \int_{\tau}^{t}\!\! \int_{I'_{1}} \bigl(1+2\delta v(x,t+\Delta t)\bigr) \bigl(k_{2}(x)w_{x(t)} \bigr)^{2}_{x} \xi^{2}\,\mathrm{d}x\,\mathrm{d} t \\& \quad \leq\vartheta_{1} \int_{\tau_{1}}^{t}\!\! \int_{I'_{1}} \bigl(k_{2}(x)w_{x(t)} \bigr)^{2}_{x}\xi^{2}\,\mathrm{d}x\,\mathrm{d} t+ \vartheta_{2} \int_{\tau_{1}}^{t}\!\! \int_{I'_{1}} w^{2}_{t(t)}\xi^{2}\, \mathrm {d}x\,\mathrm{d}t +C\bigl(\vartheta_{1},\vartheta_{2},d', \tau\bigr) J_{2}, \end{aligned}$$
(4.19)
where \(J_{2}=\int_{\tau_{1}}^{t}\int_{I'_{1}} [w^{2}_{x(t)}+v^{2}_{(t)} (k_{2}(x)w_{x} )^{2}_{x} +u^{2}_{(t)}+v^{2}_{(t)} ]\,\mathrm{d}x\,\mathrm{d}t\). Moreover, it follows from the equations in (4.18) that
$$ \int_{\tau_{1}}^{t}\!\! \int_{I'_{1}} w^{2}_{t(t)}\xi^{2}\, \mathrm{d}x\,\mathrm {d}t \leq C^{*} \int_{\tau_{1}}^{t}\!\! \int_{I'_{1}} \bigl(k_{2}(x)w_{x(t)} \bigr)^{2}_{x}\xi ^{2}\,\mathrm{d} x\, \mathrm{d}t+CJ_{2}, $$
(4.20)
and from (4.14)-(4.17) that
$$ J_{2}\leq \int_{\tau_{1}}^{t}\!\! \int_{I'_{1}} \bigl[w^{2}_{xt}+ \bigl(k_{2}(x)w_{x} \bigr)^{2}_{x} +u^{2}_{t}+1 \bigr]\,\mathrm {d}x\,\mathrm{d}t \leq C \bigl(d',\tau\bigr). $$
(4.21)
Choosing \(\vartheta_{1}\), \(\vartheta_{2}\), such that \(\vartheta _{1}+C^{*}\vartheta_{2}=1/2\), we find from (4.19)-(4.21) that
$$ \int_{I'_{1}}w^{2}_{x(t)}\xi^{2}\, \mathrm{d}x + \int_{\tau _{1}}^{t}\!\! \int_{I'_{1}} \bigl(k_{2}(x)w_{x(t)} \bigr)^{2}_{x}\xi^{2}\,\mathrm {d}x\,\mathrm{d} t \leq C\bigl(d',\tau\bigr). $$
Then
$$ \sup_{t\in[\tau_{2},T]}\bigl\Vert w_{xt}(\cdot,t)\bigr\Vert _{L^{2}(I'_{2})}\leq C\bigl(d',\tau\bigr), $$
which along with (4.17) further implies that
$$ \sup_{t\in[\tau_{2},T]}\bigl\Vert v_{xt}(\cdot,t)\bigr\Vert _{L^{2}(I'_{2})}\leq C\bigl(d',\tau\bigr). $$
(4.22)
We next use (4.22) to prove that u possesses property (ii) in Definition 2.1. It is obvious that \(u(x,t)\) is a generalized solution of the single equation \(z_{t}- (\tilde{A}(x,t,z, z_{x}) )_{x}+\check{A}(x,t,z)=0\) in the sense of Definition 1.1 in [19], where
$$\begin{aligned}& \tilde{A}= \tilde{A}(x,t,z, p):=k_{1}(x) \bigl(1+2\beta z+v(x,t) \bigr)p+k_{1}(x)zv_{x}(x,t), \\& \check{A}=\check{A}(x,t,z):=-zf_{1}\bigl(x,z,v(x,t)\bigr). \end{aligned}$$
From (4.15) and (4.16) it follows that when \((x,t,z,p)\in(I'_{2}\cap I^{(i)})\times(\tau_{2},T]\times[0,C_{1}]\times \mathbb{R}\),
$$\begin{aligned}& \bigl\vert \tilde{A}(x,t,z, p)\bigr\vert \leq C\bigl(d',\tau \bigr) \bigl(1+\vert p\vert \bigr),\qquad \mu_{0}\leq \tilde{A}_{p}\leq C, \\& \vert \tilde{A}_{z}\vert \leq C\bigl(d',\tau\bigr) \bigl(1+\vert p\vert \bigr), \qquad \vert \tilde{A}_{x}\vert \leq C \bigl(d',\tau\bigr) \bigl(1+\vert p\vert \bigr), \\& \vert \tilde{A}_{t}\vert \leq C \vert p\vert +C \phi_{1}, \qquad \bigl\vert \check{A}(x,t,z)\bigr\vert +\vert \check{A}_{z}\vert +\vert \check{A}_{x}\vert +\vert \check{A}_{t}\vert \leq C\bigl(d',\tau\bigr), \end{aligned}$$
where \(\phi_{1}:=|v_{xt}|\), and from (4.22) that
$$ \sup_{t\in[\tau_{2},T]}\bigl\Vert \phi_{1}(\cdot,t)\bigr\Vert _{L^{2}(I'_{2})}\leq C\bigl(d',\tau\bigr). $$
Consequently, the conditions of Theorem 1.1 in [19] are fulfilled. Then Theorem 1.1 in [19] shows that
$$ \|u_{t}\|_{ C^{\tilde{\alpha}'_{1}}(\bar{I}'_{3}\times[\tau_{3},T])}\leq C(d,\tau). $$
(4.23)
Since \([k_{1,i} ((1+\beta u+\gamma v)u )_{x} ]_{x}=uf_{1,i}(u,v)-u_{t}\) on \((I'_{2}\cap I^{(i)})\times (\tau_{2},T]\), then by (4.23) we get \([k_{1,i} ((1+\beta u+\gamma v)u )_{x} ]_{x}\in C^{\tilde{\alpha}'_{2} }((\bar {I}'_{3}\cap \bar{I}^{(i)})\times[\tau_{3},T])\). Thus \((1+2\beta u+\gamma v )u_{ x}+\gamma u v_{ x}\in C^{1+\tilde{\alpha}'_{2},\tilde{\alpha}'_{2} }((\bar {I}'\cap \bar{I}^{(i)})\times[\tau,T])\). In view of \(u,v\in C^{\alpha' }(\bar{I}'\times[0,T])\) and \(v_{x}\in C^{\bar{\alpha}'_{1} }((\bar {I}'\cap \bar{I}^{(i)})\times[\tau,T])\), we have \(u_{x}\in C^{\tilde{\alpha}'_{3} }((\bar{I}'\cap\bar{I}^{(i)})\times[\tau,T])\). Since \((u,v)\) satisfies pointwise the equations in (1.1) on \(Q^{(i)}_{T}\) (\(i=1,2\)), then we further use integral identities (4.1), (4.2), the regularity of \((u,v)\) in the neighborhood of \(\Gamma_{T}\) and the compatibility conditions in (2.3) to conclude that \((u,v)\) satisfies pointwise the inner boundary conditions in (1.1) on \(\Gamma_{T}\), and it satisfies homogeneous Neumann boundary conditions on \(S_{T}\) for almost all t (see Chapter III, Section 13 in [18]). □

From Lemmas 4.1-4.3 we see that problem (1.1) has at least one solution.

Lemma 4.4

The solution of problem (1.1) is unique.

Proof

Suppose that \((u_{1},v_{1})\), \((u_{2},v_{2})\) are solutions of (1.1). Let
$$ \tilde{u}=u_{1}-u_{2},\qquad \tilde{v}=v_{1}-v_{2}, \qquad \tilde{\mathbf{w}}=(\tilde{u},\tilde{v} ). $$
Set \(\eta=\tilde{v}\) in (4.2). Then by a subtraction of the two integral identities (4.2) for \(v_{1}\), \(v_{2}\), we have
$$\begin{aligned}& \frac{1}{2} \int_{0}^{l}\tilde{v}^{2}(x,t)\,\mathrm {d}x+ \int\!\!\! \int _{Q_{t}} k_{2}(x) (1+2\delta v_{1}) \tilde{v}^{2}_{x}\,\mathrm{d}x\,\mathrm{d}t \\& \quad = \int\!\!\! \int_{Q_{t}} \bigl\{ -2\delta k_{2}(x) v_{2x} \tilde{v}\tilde{v}_{x}+\tilde{v}^{2}f_{2}(x,u_{1},v_{1}) \\& \qquad {}+ v_{2} \bigl[f_{2}(x,u_{1},v_{1}) -f_{2}(x,u_{2},v_{2}) \bigr]\tilde{v} \bigr\} \, \mathrm{d}x\,\mathrm{d}t. \end{aligned}$$
Using Cauchy’s inequality we get
$$ \int_{0}^{l}\tilde{v}^{2}(x,t)\, \mathrm{d}x+ \int\!\!\! \int _{Q_{t}}\tilde {v}^{2}_{x}\,\mathrm{d}x \,\mathrm{d}t\leq C \int\!\!\! \int_{Q_{t}}\bar{\psi }^{2}\bigl(\tilde{u}^{2}+ \tilde {v}^{2}\bigr)\,\mathrm{d}x\,\mathrm{d}t, $$
(4.24)
where \(\bar{\psi}:=1+|u_{1x}|+|u_{2x}| +|v_{1x}|+|v_{2x}|\).
Similarly, set \(\eta=\tilde{u}\) in (4.1). By a subtraction of the two integral identities (4.1) for \(u_{1}\), \(u_{2}\), we obtain
$$\begin{aligned}& \frac{1}{2} \int_{0}^{l}\tilde{u}^{2}(x,t)\,\mathrm {d}x+ \int\!\!\! \int _{Q_{t}} k_{1}(x) (1+2\beta u_{1}+ \gamma v_{1})\tilde{u}^{2}_{x}\,\mathrm{d}x\, \mathrm{d}t \\& \quad =- \int\!\!\! \int_{Q_{t}} \bigl\{ \bigl[k_{1}(x) (2\beta\tilde{u}+ \gamma \tilde{v}) \bigr]u_{2x}+k_{1}(x)\gamma u_{1} v_{1x}-k_{1}(x)\gamma u_{2} v_{2x} \bigr\} \tilde{u}_{x}\,\mathrm{d}x\,\mathrm{d}t \\& \qquad {}+ \int\!\!\! \int_{Q_{t}} \bigl\{ \tilde{u}f_{1}(x,u_{1},v_{1})+ u_{2} \bigl[f_{1}(x,u_{1},v_{1}) -f_{1}(x,u_{2},v_{2}) \bigr] \bigr\} \tilde{u}\, \mathrm{d}x\,\mathrm{d}t, \end{aligned}$$
which, together with Cauchy’s inequality, implies that
$$ \int_{0}^{l}\tilde{u}^{2}(x,t)\, \mathrm{d}x+ \int\!\!\! \int _{Q_{t}}\tilde {u}^{2}_{x}\,\mathrm{d}x \,\mathrm{d}t \leq C \int\!\!\! \int_{Q_{t}}\bar{\psi}^{2}\bigl(\tilde{u}^{2}+ \tilde {v}^{2}\bigr)\,\mathrm{d}x\,\mathrm{d} t+C \int\!\!\! \int_{Q_{t}}\tilde{v}^{2}_{x}\,\mathrm{d}x\, \mathrm{d}t. $$
(4.25)
Furthermore, (4.24), (4.25) yield
$$ \int_{0}^{l} |\tilde{\mathbf{w}}|^{2}(x,t) \,\mathrm {d}x+ \int\!\!\! \int _{Q_{t}}|\tilde{\mathbf{w}}_{x}|^{2}\, \mathrm{d}x\,\mathrm{d} t\leq C \int_{0}^{t} \biggl\{ \bigl\Vert \bar{\psi}( \cdot,t)\bigr\Vert ^{2}_{L^{\infty}(0,l)} \int_{0}^{l} |\tilde{\mathbf{w} }|^{2}\, \mathrm{d} x \biggr\} \,\mathrm{d}t. $$
Again by using Gronwall’s inequality we deduce that \(\tilde{\mathbf{w}}=0\). Then the solution of (1.1) is unique. □

By Lemmas 4.1-4.4 we complete the proof of Theorem 2.1.

Declarations

Acknowledgements

The authors would like to thank the reviewers and the editors for their valuable suggestions and comments. The work was supported by the research fund of Department of Education of Sichuan Province (15ZB0343) and the research fund of Chengdu Normal University (CS14ZD01).

Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors’ Affiliations

(1)
College of Mathematics, Chengdu Normal University

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