Skip to main content

Advertisement

On an electrorheological fluid equation with orientated convection term

Abstract

A kind of electrorheological fluid equations with orientated convection terms is considered. If the diffusion coefficient \(a(x,t)\in C^{1}(\overline{Q_{T}})\) is degenerate on the boundary ∂Ω, not only the uniqueness of weak solution is proved, but also the stability of the solutions can be proved without any boundary condition, provided that there are some restrictions on the diffusion coefficient \(a(x,t)\) and the convective coefficient \(\vec{b}(x,t)\). Moreover, the large time behavior of weak solution is studied.

Introduction

The initial boundary value problem of an electrorheological fluids equation with orientated convection term

$$\begin{aligned}& u_{t} =\operatorname{div}\bigl(a(x,t) \vert \nabla u \vert ^{p(x,t)-2}\nabla u\bigr)+ \vec{f}(x,t)\cdot \nabla u^{q},\quad (x,t) \in {Q_{T}}=\varOmega \times (0,T), \end{aligned}$$
(1.1)
$$\begin{aligned}& u(x,0)= u_{0}(x),\quad x\in \varOmega , \end{aligned}$$
(1.2)
$$\begin{aligned}& u(x,t)=0,\quad (x,t)\in \partial \varOmega \times (0,T), \end{aligned}$$
(1.3)

is studied in this paper, where \(1< p(x,t)\in C(\overline{Q_{T}})\), \(q>0\), \(a(x,t) \in C^{1}(\overline{Q_{T}})\), \(\vec{f}=\{f^{i}(x,t)\}\), \(f^{i}(x,t)\in C^{1}(\overline{Q_{T}})\), and \(\varOmega \subset \mathbb{R}^{N}\) is a bounded domain with a smooth boundary ∂Ω.

When \(p(x,t)>1\) is a measurable function on \(Q_{T}\), equation (1.1) arises in electrorheological fluids theory [1]. If \(\vec{f}(x,t)=0\), \(a(x,t)=1\) for all \((x,t)\in \overline{Q_{T}}\), the existence and uniqueness results of equation (1.1) have been obtained in [2,3,4,5,6] etc. If \(p(x,t)=p>1\) is a constant, \(a(x,t)=1\) and \(\vec{f}(x,t)=0\), equation (1.1) is well known as non-Newtonian fluid equation and has been studied by many mathematicians, one can refer to [7,8,9,10,11,12,13,14,15] and the references therein. From these papers, we know that the uniqueness and the stability of weak solutions can be proved if the Dirichlet boundary value condition (1.3) is imposed. In recent years, the equations with the type

$$ u_{t} =\operatorname{div}\bigl(a(x,t) \vert \nabla u \vert ^{p(x,t)-2}\nabla u\bigr)+f(x,t,u, \nabla u),\quad (x,t) \in {Q_{T}} $$
(1.4)

have drawn wide public attention [16,17,18,19,20,21,22,23] etc. When \(p(x,t)=p>1\) is a constant, \(f(x,t,u,\nabla u)\) is a linear function, the well-posedness problem of equation (1.4) was studied in [24,25,26]. In addition, the non-Newtonian polytropic filtration equation with orientated convection

$$ u_{t} =\operatorname{div}\bigl( \bigl\vert \nabla u^{m} \bigr\vert ^{p-2}\nabla u^{m}\bigr)+ \vec{b}(x) \cdot \nabla u^{q},\quad x\in \mathbb{R}^{N}, t>0, $$

was studied in [27], where \(m>0\), \(p>2\) and \(\vec{b}=\{b^{i}(x) \}\), \(b_{i}(x)\in C^{1}(\mathbb{R})^{N}\). The author has been interested in the stability of weak solutions to equation (1.4) for a long time. When \(a(x,t)=a(x)\), \(a(x)> 0\) in Ω and

$$ a(x)=0,\quad x\in \partial \varOmega , $$
(1.5)

some progresses have been made in [22, 23]. If \(a(x,t)=a(x)\), \(p(x,t)=p\), the stability of weak solutions to equation (1.1) has been studied in [28, 29]. We have found that condition (1.5) may replace the usual Dirichlet boundary value condition (1.3) for some special \(f(x,t,u, \nabla u)\), the stability of solutions can be established without any boundary value condition (1.3), provided that there are some other restrictions on \(f(x,t,u,\nabla u)\).

In this paper, we first generalize the results contained in [15, 28, 29] to equation (1.1), since there is time variable t in the exponents, there are some essential difficulties that should be overcome. Secondly, we will used some ideas [3, 4, 30] to prove the uniqueness of weak solution. Thirdly, the large time behavior of weak solutions is studied free from the limitations of the boundary value condition.

We denote that

$$ p_{+}=\max_{(x,t)\in \overline{Q_{T}}}p(x,t),\qquad p_{-}= \min_{(x,t)\in \overline{Q_{T}}}p(x,t), $$

assume that \(p_{-}>1\), and the constants c appearing in different places represent different constants, \(a(x,t)\) is a nonnegative function in \(C^{1}(\overline{Q_{T}})\), and for every \(t\in [0,T]\),

$$ a(x,t)=0, \quad x\in \partial \varOmega \quad \text{and}\quad a(x,t)>0, \quad x \in \varOmega . $$
(1.6)

We give the basic definitions and the main results now.

Definition 1.1

If a nonnegative function \(u(x,t)\) satisfies

$$ u \in L^{\infty }(Q_{T}), \qquad u_{t}\in L^{{p_{+}}'}\bigl(0,T;W^{-1,{p_{+}}'}(\varOmega )\bigr),\qquad a(x,t){ \vert { \nabla u} \vert ^{p(x,t)}} \in {L^{\infty }}\bigl(0,T;L^{1}( \varOmega )\bigr), $$

and for any \(\varphi (x,t) \in C_{0}^{1}(\overline{Q_{T}})\),

$$ \begin{gathered} \iint _{{Q_{T}}} \biggl[\frac{\partial u}{\partial t}\varphi + a(x,t) \vert \nabla u \vert ^{p(x,t) - 2}\nabla u \cdot \nabla \varphi \biggr] \,dx \,dt \\ \qquad {}+\sum_{i=1}^{N} \iint _{{Q_{T}}}u^{q} \bigl[f^{i}_{x_{i}}(x,t) \varphi +f^{i}(x,t)\varphi _{x_{i}} \bigr] \,dx \,dt \hfill \\ \quad = 0, \end{gathered} $$

then we say \(u(x,t)\) is a solution of equation (1.1) with the initial value (1.2) which is satisfied in the sense

$$ \lim_{t\rightarrow 0} \int _{\varOmega }u(x,t)\phi (x)\,dx= \int _{\varOmega } u _{0}(x) \phi (x)\,dx $$
(1.7)

for any \(\phi (x)\in C_{0}^{\infty }(\varOmega )\).

Here, \(p'_{+}=\frac{p_{+}}{p_{+}-1}\), \(b^{i}_{x_{i}}=\frac{\partial b ^{i}(x)}{\partial x_{i}}\), \(g_{x_{i}}=\frac{\partial g}{\partial x_{i}}\) as usual, \(i=1,2,\ldots , N\). In this paper, the existence of the nonnegative solution is proved.

Theorem 1.2

If \(p_{-}>1\), \(1\leq q< p_{+}\), \(a(x,t)\geq 0\) satisfies (1.6),

$$ 0\leq u_{0} \in L^{\infty }(\varOmega ),\qquad a(x,0) \vert \nabla u_{0} \vert ^{p(x,0)} \in L^{1}(\varOmega ),\quad i=1,2,\ldots , N, $$
(1.8)

then equation (1.1) with initial value (1.2) has a nonnegative weak solution u.

If \(\int _{\varOmega }a(x,t)^{-\frac{1}{p(x,t)-1}}\,dx<\infty \) for any \(t\in [0,T]\), similar as the proof of Theorem 1.1 in [12], the weak solution u in Theorem 1.2 satisfies

$$ \begin{aligned}[b] & \int _{\varOmega } \vert \nabla u \vert \,dx \\ &\quad = \int _{ \{ x\in \varOmega :a(x,t)^{\frac{1}{p(x,t)-1}} \vert \nabla u \vert \leq 1 \}} \vert \nabla u \vert \,dx+ \int _{ \{ x\in \varOmega :a(x,t)^{\frac{1}{p(x,t)-1}} \vert \nabla u \vert > 1 \}} \vert \nabla u \vert \,dx \\ &\quad \leq \int _{\varOmega } a(x,t)^{-\frac{1}{p(x,t)-1}} \,dx+ \int _{\varOmega } a(x,t) \vert \nabla u \vert ^{p(x,t)} \,dx \\ &\quad \leq c . \end{aligned} $$
(1.9)

Then the boundary value condition (1.3) is valid in the sense of the trace. However, in general, \(u(x,t)\) is in \(W_{\mathrm{loc}}^{1,p(x,t)}( \varOmega )\) and cannot be defined the trace on the boundary. Accordingly, instead of considering the boundary value condition itself, we would pay a close attention to finding some other conditions to replace the boundary value condition and prove the corresponding stability of weak solutions (or uniqueness of weak solution).

Theorem 1.3

Let \(q\geq 1\), \(a(x,t)\geq 0\) satisfy (1.6), \(p(x,t)\geq p_{-}>1\), \(u(x,t)\) and \(v(x,t)\) be two nonnegative weak solutions of equation (1.1) with the initial values \(u_{0}(x)\) and \(v_{0}(x)\). If

$$\begin{aligned}& \bigl\vert f^{i}(x,t) \bigr\vert \geq ca(x,t), \quad i=1,2, \ldots , N, t\in [0, T], \end{aligned}$$
(1.10)
$$\begin{aligned}& \int _{\varOmega }a(x,t)^{-(p(x,t)-1)}\,dx< \infty ,\quad t\in [0, T], \end{aligned}$$
(1.11)

then

$$ \int _{\varOmega } \bigl\vert u(x,t) - v(x,t) \bigr\vert \,dx \leq c \int _{ \varOmega } \bigl\vert u_{0}(x) - v_{0}(x) \bigr\vert \,dx,\quad \textit{a.e. }t\in [0,T). $$
(1.12)

Theorem 1.4

If \(q\geq 1\), \(a(x,t)\geq 0\) satisfies (1.6), \(p(x,t)\geq p_{-}>1\), \(u(x,t)\) and \(v(x,t)\) are two nonnegative weak solutions of equation (1.1) with the initial values \(u_{0}(x)=v_{0}(x)\),

$$\begin{aligned}& \int _{\varOmega }a(x,t)^{-\frac{1}{p(x)-1}}\,dx< \infty ,\quad t\in [0,T], \end{aligned}$$
(1.13)
$$\begin{aligned}& \operatorname{div} \vec{f}(x,t)\geq 0, \end{aligned}$$
(1.14)

then

$$ u(x,t)=v(x,t),\quad (x,t)\in Q_{T}. $$
(1.15)

One can see that condition (1.11) in Theorem 1.3 and condition (1.13) in Theorem 1.4 are complementary to each other. In this paper, u represents the gradient of u on the spatial variable x, \(\operatorname{div} \vec{f}(x,t)\) represents the divergence of f⃗ on the spatial variable x.

By the uniqueness of weak solutions, we will study the large time behavior of weak solutions without the boundary value condition in the last section.

The existence of weak solutions

In this section, we use the parabolically regularized method to prove Theorem 1.2. Consider the initial boundary value problem

$$\begin{aligned}& \begin{aligned}[b] &u_{\varepsilon t} - \varepsilon \operatorname{div}\bigl( \vert \nabla u_{\varepsilon } \vert ^{p _{+}-2}\nabla u_{\varepsilon }\bigr) -\operatorname{div}\bigl( a(x,t) \vert \nabla {u_{\varepsilon }} \vert ^{p(x,t)- 2}\nabla u_{\varepsilon }\bigr) - \vec{f}(x,t)\cdot \nabla u_{\varepsilon }^{q} \\ &\quad = 0,\quad (x,t)\in {Q_{T}}, \end{aligned} \end{aligned}$$
(2.1)
$$\begin{aligned}& {u_{\varepsilon }}(x,t) = 0, \quad (x,t) \in \partial \varOmega \times (0,T), \end{aligned}$$
(2.2)
$$\begin{aligned}& {u_{\varepsilon }}(x,0) = {u_{\varepsilon ,0}}(x), \quad x \in \varOmega , \end{aligned}$$
(2.3)

where \(0\leq u_{\varepsilon ,0} \in C^{\infty }_{0}(\varOmega )\), \(\|u_{\varepsilon ,0}\|_{L^{\infty }(\varOmega )}\leq \|u_{0}\|_{L^{ \infty }(\varOmega )}\), \(u_{\varepsilon ,0}\rightarrow u_{0}(x)\) in \(W_{0}^{1,{p_{+}}}(\varOmega )\). Then there is a unique nonnegative solution \(u_{\varepsilon }\in L^{p_{+}}(0,T; W_{0}^{1,p _{+}}(\varOmega ))\) [5], which satisfies

$$ \Vert u_{\varepsilon } \Vert _{L^{\infty }(Q_{T})} \leqslant c. $$
(2.4)

By multiplying (2.1) with \(u_{\varepsilon }\), integrating it over \(Q_{t}=\varOmega \times [0,t]\), we achieve

$$ \begin{aligned}[b] &\frac{1}{2} \int _{\varOmega }u_{\varepsilon }^{2}(x,t)\,dx+ \frac{\varepsilon }{2} \iint _{{Q_{t}}} \vert \nabla u_{\varepsilon } \vert ^{p_{+}} \,dx \,dt\\ &\quad {} + \iint _{ {Q_{t}}} a(x,t) \vert \nabla u_{\varepsilon } \vert ^{p(x,t)} \,dx \,dt \leqslant c, \quad \forall t\in [0,T). \end{aligned} $$
(2.5)

Here, we have used the following fact:

$$ \begin{aligned} \iint _{{Q_{T}}} \bigl\vert u_{\varepsilon }\vec{f}(x,t) \cdot \nabla u_{\varepsilon }^{q} \bigr\vert \,dx \,dt&=q \iint _{{Q_{T}}} \bigl\vert u^{q}_{\varepsilon } \vec{f}(x,t)\cdot \nabla u_{\varepsilon } \bigr\vert \\ & \leq \frac{\varepsilon }{2} \iint _{{Q_{T}}} \vert \nabla u_{\varepsilon } \vert ^{p _{+}} \,dx \,dt+c(\varepsilon ). \end{aligned} $$

By (2.5), we achieve

$$ \iint _{Q_{T}}a(x,t) \vert \nabla u_{\varepsilon } \vert ^{p(x,t)} \,dx \,dt\leq c $$
(2.6)

and

$$ \varepsilon \iint _{Q_{T}} \vert \nabla u_{\varepsilon } \vert ^{p_{+}} \,dx \,dt\leq c. $$
(2.7)

Let \(v\in L^{p_{+}}(0,T; W^{1,p_{+}}_{0}(\varOmega ))\), \(\|v\|_{L^{p_{+}}(0,T; W^{1,p_{+}}_{0}(\varOmega ))}=1\). Since \(u_{\varepsilon }\in L^{p_{+}}(0,T; W^{1,p_{+}}_{0}(\varOmega ))\cap L ^{\infty }(Q_{T})\), we have

$$ \begin{aligned}[b] &\langle u_{\varepsilon t}, v \rangle+\varepsilon \iint _{Q_{T}} \vert \nabla u_{\varepsilon } \vert ^{p_{+}-2}\nabla u_{\varepsilon }\nabla v \,dx \,dt \\ &\qquad {}+ \iint _{Q_{T}}a(x,t) \vert \nabla u_{\varepsilon } \vert ^{p(x,t)-2}\nabla u _{\varepsilon }\nabla v \,dx \,dt \\ &\qquad {}+\sum_{i=1}^{N} \iint _{Q_{T}}u^{q} \bigl[f^{i}_{x_{i}}(x,t)v+f^{i}(x,t)v _{x_{i}} \bigr] \,dx \,dt \\ &\quad =0. \end{aligned} $$
(2.8)

By Young’s inequality, we extrapolate that

$$ \Vert u_{\varepsilon t} \Vert _{L^{{p'}_{+}}(0,T;W^{-1,{p'}_{+}}(\varOmega ))} \leq c, $$
(2.9)

where \({p'}_{+}=\frac{p_{+}}{p_{+}-1}\) as before.

For any \(\phi \in C_{0}^{1}(\varOmega )\), \(0\leq \phi \leq 1\), it is not difficult to show that

$$ \bigl\Vert (\phi u_{\varepsilon })_{t} \bigr\Vert _{L^{{p'}_{+}}(0,T;W^{-1,{{p'}_{+}}}( \varOmega ))}\leq c $$
(2.10)

by (2.9).

Since \(H_{0}^{s}(\varOmega )\hookrightarrow W^{1, p_{+}}(\varOmega )\) when \(s>\frac{N}{2}+1\), we have \(W^{-1,{p'_{+}}}(\varOmega )\hookrightarrow H ^{-s}(\varOmega )\). Then

$$ \bigl\Vert (\phi u_{\varepsilon })_{t} \bigr\Vert _{L^{p'_{+}}(0,T;H^{-s}(\varOmega ))} \leq c. $$
(2.11)

In addition, we have

$$ \iint _{Q_{T}} \bigl\vert \nabla (\phi u_{\varepsilon }) \bigr\vert ^{p_{-}} \,dx \,dt\leq c( \phi ) \biggl(1+ \int _{0}^{T} \int _{\varOmega _{\phi }} \vert \nabla u_{\varepsilon } \vert ^{p _{-}} \,dx \,dt\biggr)\leq c(\phi ), $$

where \(\varOmega _{\phi }=\operatorname{supp}\phi \). Thus,

$$ \Vert \phi u_{\varepsilon } \Vert _{L^{p'_{+}}(0,T;W^{1,p_{-}}_{0}(\varOmega ))} \leq c. $$
(2.12)

Since \(W^{1,p_{-}}_{0}(\varOmega )\hookrightarrow L^{p_{-}}(\varOmega ) \hookrightarrow H^{-s}(\varOmega )\), Aubin’s compactness theorem in [33] yields \(\phi u_{\varepsilon }\rightarrow \phi u\) strongly in \(L^{{p'}_{+}}(0,T;L^{p_{-}}(\varOmega ))\). Then \(\phi u_{\varepsilon } \rightarrow \phi u\) a.e. in \(Q_{T}\), and so \(u_{\varepsilon }\rightarrow u\) a.e. in \(Q_{T}\).

Combining (2.4), (2.5), (2.6), and (2.7), there exist a function u and an n-dimensional vector function \(\overrightarrow{\zeta }= ({\zeta _{1}}, \ldots ,{\zeta _{n}})\) such that

$$\begin{aligned}& \varepsilon \vert \nabla u_{\varepsilon } \vert ^{p_{+}-2}\nabla u_{\varepsilon }\rightharpoonup 0, \quad \text{in } L^{\frac{p_{+}}{p_{+}-1}}(Q_{T}), \\& u \in L^{\infty }(Q_{T}),\quad \vert \zeta _{i} \vert \in L^{1}\bigl(0 T; L^{\frac{p(x,t)}{p(x,t) - 1}}(\varOmega )\bigr), \end{aligned}$$

and

$$\begin{aligned}& {u_{\varepsilon }} \rightharpoonup u, \quad \text{weakly star in } {L^{\infty }(Q_{T})},\qquad u_{\varepsilon }\rightarrow u,\quad \text{a.e. in } Q_{T}, \\& u_{\varepsilon }^{q}\rightarrow u^{q},\quad \text{a.e. in } Q_{T}, \\& {u_{\varepsilon x_{i}}} \rightharpoonup u_{x_{i}} \quad \text{in } L^{1}\bigl(0,T; L_{\mathrm{loc}}^{p(x,t)}(\varOmega )\bigr), \\& a(x,t) \vert \nabla u_{\varepsilon } \vert ^{p(x,t)- 2} \nabla u _{\varepsilon } \rightharpoonup \overrightarrow{\zeta }, \quad \text{in }\bigl\{ L^{1}\bigl(0,T; L^{\frac{p(x,t)}{p(x,t)-1}}(\varOmega )\bigr)\bigr\} ^{N}. \end{aligned}$$

Meanwhile, similar as the proof of Lemma 2.6 in [4], we can prove that

$$ \iint _{Q_{T}} a(x,t) \vert \nabla u \vert ^{p(x,t) - 2}\nabla u \cdot \nabla \varphi \,dx \,dt = \iint _{Q_{T}} \overrightarrow{\zeta } \cdot \nabla \varphi \,dx \,dt $$

for any given function \(\varphi \in C_{0}^{1} ({Q_{T}})\). Then, for any \(\varphi \in C_{0}^{1}(Q_{T})\),

$$ \begin{aligned}[b] &\langle u_{t},\varphi \rangle+ \iint _{{Q_{T}}} \Biggl[a(x,t) \vert \nabla u \vert ^{p(x,t)- 2} \nabla u \nabla \varphi +\sum_{i=1}^{N}u^{q} \bigl(f^{i}(x,t) \varphi _{x_{i}}+f^{i}_{x_{i}}(x,t) \varphi \bigr) \Biggr] \,dx \,dt \\ &\quad = 0. \end{aligned} $$

Moreover, one can prove the initial value condition in the sense of (1.7) as in [2], thus u is a solution of equation (1.1) with the initial value (1.2) in the sense of Definition 1.1. The proof is complete.

The proof of Theorem 1.3

The following lemma can be found in [30, 31].

Lemma 3.1

The variable exponent spaces \(L^{p(x)}(\varOmega )\), \(W^{1,p(x)}(\varOmega )\), and \(W^{1,p(x)}_{0}(\varOmega )\) are reflexive Banach spaces. The following hold:

  1. (i)

    Let \(p_{1}(x)\) and \(p_{2}(x)\) be real functions with \(\frac{1}{p _{1}(x)}+\frac{1}{p_{2}(x)} = 1\). Then the conjugate space of \(L^{p_{1}(x)}(\varOmega )\) is \(L^{p_{2}(x)}(\varOmega )\). For any \(u \in L^{p_{1}(x)}(\varOmega )\) and \(v \in L^{p_{2}(x)}(\varOmega )\), there holds

    $$ \biggl\vert \int _{\varOmega }uv \,dx \biggr\vert \leq 2 \Vert u \Vert _{L^{p_{1}(x)}(\varOmega )} \Vert v \Vert _{L^{p_{2}(x)}(\varOmega )}. $$
  2. (ii)

    Let \(p_{1+}=\max_{x\in \overline{\varOmega }}p_{1}(x)\), \(p_{1-}= \min_{x\in \overline{\varOmega }}p_{1}(x)\).

    $$ \begin{aligned}[b] & \textit{If } \Vert u \Vert _{L^{p_{1}(x)}(\varOmega )} = 1,\quad \textit{then } \int _{\varOmega } \vert u \vert ^{p_{1}(x)} \,dx = 1. \\ &\textit{If } \Vert u \Vert _{L^{p_{1}(x)}}(\varOmega ) > 1, \quad \textit{then } \vert u \vert ^{p_{1-}}_{L^{p_{1}(x)}} \leq \int _{\varOmega } \vert u \vert ^{p_{1}(x)} \,dx\leq \vert u \vert ^{p_{1+}}_{L^{p_{1}(x)}}. \\ &\textit{If } \Vert u \Vert _{L^{p_{1}(x)}}(\varOmega ) < 1,\quad \textit{then } \vert u \vert ^{p_{1+}}_{L^{p_{1}(x)}} \leq \int _{\varOmega } \vert u \vert ^{p_{1}(x)} \,dx\leq \vert u \vert ^{p_{1-}}_{L^{p_{1}(x)}}. \end{aligned} $$

Lemma 3.2

(see [2])

Let \(v\in L^{p_{+}}(0,T;W_{0}^{1,p _{+}}(\varOmega ))\), \(v_{t}\in L^{p_{+}'}(0,T;W^{-1,{p_{+}}'}(\varOmega ))\). For any continuous function \(h(s)\), \(H(s)=\int _{0}^{s}h(s)\,ds\), a.e. \(t_{1}, t_{2}\in [0, T)\),

$$ \int _{t_{1}}^{t_{2}}\bigl\langle v_{t}, h(v)\bigr\rangle \,dt= \int _{\varOmega }\bigl(H(v) (x,t_{2})-H(v) (x,t _{1})\bigr)\,dx. $$
(3.1)

For small \(r>0\), let

$$ S_{\sigma }(s)= \int _{0}^{s}h_{\sigma }(\tau )\,d\tau ,\qquad h_{\sigma }(s)=\frac{2}{ \sigma } \biggl(1-\frac{\mid s\mid }{\sigma } \biggr)_{+}. $$

Then

$$ \lim_{\sigma \rightarrow 0}S_{\sigma }(s)= \operatorname{sgn}s, \qquad \lim _{ \sigma \rightarrow 0}H_{\sigma }(s)= \int _{0}^{s}S_{\sigma }( \tau )\,d\tau = \vert s \vert $$
(3.2)

and

$$ \lim _{\sigma \rightarrow 0}sh_{\sigma }(s)=0. $$
(3.3)

Let \(\varphi (x,t)\) be a nonnegative function in \(C^{1}(\overline{Q _{T}})\), and for every \(t\in [0,T]\),

$$ \varphi (x,t)=0,\quad x\in \partial \varOmega \quad \text{and}\quad \varphi (x,t)>0,\quad x\in \varOmega . $$
(3.4)

Theorem 3.3

If \(q\geq 1\), \(a(x,t)\geq 0\) satisfies (1.6), \(p(x,t)\geq p_{-}>1\), \(u(x,t)\) and \(v(x,t)\) are two nonnegative weak solutions of equation (1.1) with the initial values \(u_{0}(x)\) and \(v_{0}(x)\) respectively, and there is a nonnegative function \(\varphi \in C^{1}(\overline{Q_{T}})\) satisfying (3.4) such that

$$\begin{aligned}& \int _{\varOmega }a(x,t) \biggl\vert \frac{\nabla \varphi }{\varphi } \biggr\vert ^{p(x,t)}\,dx< \infty , \qquad \int _{\varOmega }\frac{ \vert \sum_{i=1}^{N} f^{i}(x,t) \varphi _{x_{i}} \vert }{\varphi }\,dx< \infty ,\quad t\in [0,T], \end{aligned}$$
(3.5)
$$\begin{aligned}& \int _{\varOmega }a(x,t)^{-\frac{1}{p(x,t)-1}} \Biggl\vert \sum _{i=1}^{N}f^{i}(x,t) \Biggr\vert ^{\frac{p(x,t)}{p(x,t)-1}}\,dx< \infty ,\quad t\in [0,T], i=1,2, \ldots, N, \end{aligned}$$
(3.6)

then

$$ \int _{\varOmega } \bigl\vert u(x,t) - v(x,t) \bigr\vert \,dx \leq c \int _{ \varOmega } \bigl\vert u_{0}(x) - v_{0}(x) \bigr\vert \,dx,\quad \textit{a.e. }t\in [0,T). $$
(3.7)

Proof

For two solutions \(u(x,t)\), \(v(x,t)\), the test function can be chosen as \(S_{\sigma }(\varphi (u-v))\), where \(\varphi (x,t)\) satisfies (3.4). Then

$$\begin{aligned}& \int _{0}^{t} \int _{\varOmega }S_{\sigma }\bigl(\varphi (u-v)\bigr) \frac{\partial (u-v)}{ \partial t} \,dx \,dt \\& \qquad {}+ \int _{0}^{t} \int _{\varOmega } a(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)- 2}\nabla u - \vert \nabla v \vert ^{p(x,t)- 2}\nabla v \bigr) \cdot \nabla (u - v)S'_{\sigma } \bigl(\varphi (u-v)\bigr) \,dx \,dt \\& \qquad {}+ \int _{0}^{t} \int _{\varOmega } a(x,t) \bigl( \vert \nabla u \vert ^{p(x,t) - 2}\nabla u - \vert \nabla v \vert ^{p(x,t)- 2}\nabla v \bigr) \cdot \nabla \varphi (u - v)S'_{\sigma }\bigl( \varphi (u-v)\bigr) \,dx \,dt \\& \qquad {}+ \sum_{i=1}^{N} \int _{0}^{t} \int _{\varOmega }f^{i}_{x_{i}}(x,t) \bigl(u^{q}-v ^{q}\bigr)S_{\sigma }\bigl(\varphi (u-v)\bigr) \,dx \,dt \\& \qquad {}+\sum_{i=1}^{N} \int _{0}^{t} \int _{\varOmega }\varphi f^{i}(x,t) \bigl(u^{q} - v ^{q}\bigr)\cdot (u- v)_{x_{i}}S'_{\sigma } \bigl(\varphi (u-v)\bigr)\,dx \,dt \\& \qquad {}+ \sum_{i=1}^{N} \int _{0}^{t} \int _{\varOmega }f^{i}(x,t) \bigl(u^{q}-v^{q} \bigr) \cdot \varphi _{x_{i}} (u - v)S'_{\sigma } \bigl(\varphi (u-v)\bigr) \,dx \,dt \\& \quad =0 . \end{aligned}$$
(3.8)

Since \(a(x,t)\geq 0\), \(S'_{\sigma }(s)\geq 0\), obviously,

$$ \int _{\varOmega } a(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)- 2}\nabla u - \vert \nabla v \vert ^{p(x,t) - 2}\nabla v \bigr) \cdot \nabla (u - v)S'_{ \sigma }\bigl(\varphi (u - v)\bigr) \,dx \geq 0. $$
(3.9)

Since \(\int _{\varOmega }a(x,t) \vert \frac{\nabla \varphi }{\varphi } \vert ^{p(x,t)}\,dx<\infty \),

$$ \begin{gathered}[b] \biggl\vert \int _{\varOmega } a(x,t) (u - v)S'_{\sigma } \bigl(\varphi (u - v)\bigr) \bigl( \vert \nabla u \vert ^{p(x,t)- 2} \nabla u - \vert \nabla v \vert ^{p(x,t) - 2}\nabla v\bigr) \nabla \varphi \,dx \biggr\vert \\ \quad =\left \vert \int _{\varOmega } a(x,t)^{-\frac{p(x,t)-1}{p(x,t)}}a(x,t) (u - v)S'_{ \sigma }\bigl(\varphi (u - v)\bigr)\right . \\ \qquad {}\cdot \left .a(x,t)^{\frac{p(x,t)-1}{p(x,t)}}\bigl( \vert \nabla u \vert ^{p(x,t)- 2}\nabla u - \vert \nabla v \vert ^{p(x,t) - 2}\nabla v \bigr) \nabla \varphi \,dx\right \vert \\ \quad \leq \biggl( \int _{\varOmega } \biggl\vert a(x,t)^{\frac{1}{p(x,t)}} \frac{ \nabla \varphi }{\varphi } \varphi (u - v)S'_{\sigma }\bigl( \varphi (u - v)\bigr) \biggr\vert ^{p(x,t)}\,dx \biggr)^{\frac{1}{p_{1}(t)}} \\ \qquad {}\cdot \biggl( \int _{\varOmega }a(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)}+ \vert \nabla v \vert ^{p(x,t)}\bigr)\,dx \biggr) ^{\frac{1}{p_{1}'(t)}} \\ \quad \rightarrow 0 \end{gathered} $$
(3.10)

as \(\sigma \rightarrow 0\), where \(p'(x,t)=\frac{p(x,t)}{p(x,t)-1}\). \(p_{1}(t)=\max_{x\in \overline{\varOmega }}p(x,t)\), or \(p_{1}(t)= \min_{x\in \overline{\varOmega }}p(x,t)\) according to

$$ \int _{\varOmega } \biggl\vert a(x,t)^{\frac{1}{p(x,t)}} \frac{\nabla \varphi }{ \varphi } \varphi (u - v)S'_{\sigma }\bigl( \varphi (u - v)\bigr) \biggr\vert ^{p(x,t)}\,dx \geq 1 $$

or

$$ \int _{\varOmega } \biggl\vert a(x,t)^{\frac{1}{p(x,t)}} \frac{\nabla \varphi }{ \varphi } \varphi (u - v)S'_{\sigma }\bigl( \varphi (u - v)\bigr) \biggr\vert ^{p(x,t)}\,dx< 1 $$

by Lemma 3.1. \(p'_{1}(t)\) has a similar meaning.

In addition, since \(u,v\in L^{\infty }(Q_{T})\) and \(\int _{\varOmega }\frac{| \sum_{i=1}^{N}b^{i}(x,t)\varphi _{x_{i}}|}{ \varphi }\,dx<\infty \), the dominated convergence theorem yields

$$ \begin{gathered}[b] \lim_{\sigma \rightarrow 0} \Biggl\vert \int _{\varOmega } \sum_{i=1}^{N}f^{i}(x,t) \bigl(u ^{q} - v^{q}\bigr) \cdot \varphi _{x_{i}} (u-v) S'_{\sigma }\bigl(\varphi (u-v) \bigr) \,dx \Biggr\vert \\ \quad \leq \lim_{\sigma \rightarrow 0} \int _{\varOmega } \Biggl\vert \sum_{i=1}^{N} f ^{i}(x,t) \bigl(u^{q} - v^{q} \bigr)S'_{\sigma }\bigl(\varphi (u - v)\bigr)\varphi (u-v) \frac{ \varphi _{{x_{i}}}}{\varphi } \Biggr\vert \,dx \\ \quad \leq c\lim_{\sigma \rightarrow 0} \int _{\varOmega } \bigl\vert S'_{\sigma } \bigl(\varphi (u - v)\bigr)\varphi (u-v) \bigr\vert \frac{ \vert \sum_{i=1}^{N}f^{i}(x,t)\varphi _{x_{i}} \vert }{ \varphi }\,dx \\ \quad =0. \end{gathered} $$
(3.11)

Since

$$ \bigl\vert u^{q} - v^{q} \bigr\vert \leq c \vert u-v \vert ,\quad q\geq 1, $$

and by (3.6)

$$ \int _{\varOmega }a(x,t)^{-\frac{1}{p(x,t)-1}} \Biggl\vert \sum _{i=1}^{N}f^{i}(x,t) \Biggr\vert ^{\frac{p(x,t)}{p(x,t)-1}}\,dx\leq c, $$

we have

$$\begin{aligned}& \lim_{\sigma \rightarrow 0} \Biggl\vert \int _{{\varOmega }} \sum_{i=1}^{N}f ^{i}(x,t)\varphi \bigl(u^{q} - v^{q} \bigr){S_{\sigma }}'\bigl(\varphi (u - v)\bigr) (u - v)_{ {x_{i}}}\,dx \Biggr\vert \\& \quad \leq c \lim_{\sigma \rightarrow 0} \biggl( \int _{\varOmega }a(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)}+ \vert \nabla v \vert ^{p(x,t)}\bigr)\,dx \biggr)^{\frac{1}{p_{1}(t)}} \\& \qquad{} \cdot \Biggl( \int _{\varOmega } \Biggl\vert \sum_{i=1}^{N}a(x,t)^{- \frac{1}{p(x,t)}}f^{i}(x,t) \Biggr\vert ^{\frac{p(x,t)}{p(x,t)-1}} \bigl\vert \varphi (u-v){S_{\sigma }}' \bigl(\varphi (u - v)\bigr) \bigr\vert ^{\frac{p(x,t)}{p(x,t)-1}}\,dx \Biggr) ^{\frac{1}{p'_{1}(t)}} \\& \quad \leq c \lim_{\sigma \rightarrow 0} \Biggl( \int _{\varOmega }a(x,t)^{- \frac{1}{p(x,t)}} \Biggl\vert \sum _{i=1}^{N}f^{i}(x,t) \Biggr\vert ^{ \frac{p(x,t)}{p(x,t)-1}} \bigl\vert \varphi (u-v){S_{\sigma }}' \bigl(\varphi (u - v)\bigr) \bigr\vert ^{ \frac{p(x,t)}{p(x,t)-1}}\,dx \Biggr)^{\frac{1}{p'_{1}(t)}} \\& \quad =0. \end{aligned}$$
(3.12)

Here \(p_{1}(t)=\max_{x\in \overline{\varOmega }}p(x,t)\) or \(p_{1}(t)= \min_{x\in \overline{\varOmega }}p(x,t)\) according to

$$ \int _{\varOmega }a(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)}+ \vert \nabla v \vert ^{p(x,t)}\bigr)\,dx\geq 1 $$

or

$$ \int _{\varOmega }a(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)}+ \vert \nabla v \vert ^{p(x,t)}\bigr)\,dx< 1 $$

by Lemma 3.1 for any \(t\in [0,T)\), \(p'_{1}(t)\) has a similar meaning.

At the same time,

$$ \begin{aligned}[b] & \lim_{\sigma \rightarrow 0} \Biggl\vert \int _{\varOmega } \sum_{i=1}^{N}f^{i} _{x_{i}}(x,t) \bigl(u^{q}-v^{q} \bigr)S_{\sigma }\bigl(\varphi (u-v)\bigr)\,dx \Biggr\vert \\ &\quad \leq c \int _{\varOmega } \vert u-v \vert \,dx,\quad i=1,2,\ldots , N, \end{aligned} $$
(3.13)

is obvious by the assumption that \(f^{i}(x,t)\in C^{1}(\overline{Q _{T}})\), \(i=1,2,\ldots , N\).

By the definition of the weak characteristic function \(\varphi (x,t)\), we can employ Lemma 3.2 to deduce that

$$\begin{aligned} &\lim_{\sigma \rightarrow 0} \int _{0}^{t} \int _{\varOmega } S_{\sigma }\bigl( \varphi (u - v)\bigr) \frac{\partial (u - v)}{\partial t} \,dx \,dt \\ &\quad = \int _{\varOmega } \int _{0}^{t}\operatorname{sign}(u - v) \frac{\partial (u - v)}{ \partial t} \,dx \,dt \\ &\quad = \lim_{\sigma \rightarrow 0} \int _{0}^{t} \int _{\varOmega } S_{\sigma }(u - v)\frac{\partial (u - v)}{\partial t} \,dx \,dt \\ &\quad = \lim_{\sigma \rightarrow 0} \int _{0}^{t}(H_{\sigma }(u-v) (t)-H_{ \sigma }(u_{0}-v_{0})\,dt \\ &\quad = \int _{\varOmega } \bigl\vert u(x,t)-v(x,t) \bigr\vert \,dx- \int _{\varOmega } \bigl\vert u_{0}(x)-v_{0}(x) \bigr\vert \,dx. \end{aligned}$$
(3.14)

Let \(\sigma \rightarrow 0\) in (3.8). By (3.9), (3.11), (3.12), (3.13), and (3.14), we have

$$ \int _{\varOmega } \bigl\vert u(x,t) - v(x,t) \bigr\vert \,dx \leqslant c \int _{ \varOmega } \bigl\vert u_{0}(x)- v_{0}(x) \bigr\vert \,dx,\quad \forall t \in [0,T). $$

 □

Proof of Theorem 1.3

By conditions (1.10) and (1.11), only if we choose \(\phi (x,t)=a(x,t)\), we know conditions (4.1) (4.2) in Theorem 3.3 are true, the conclusion follows easily. □

The proof of Theorem 1.4

Theorem 4.1

If \(q\geq 1\), \(a(x,t)\geq 0\) satisfies (1.6), \(p(x,t)\geq p_{-}>1\), \(u(x,t)\) and \(v(x,t)\) are two nonnegative weak solutions of equation (1.1) and with the same initial value \(u_{0}(x)=v_{0}(x)\), and

$$\begin{aligned}& \int _{\varOmega } \Biggl\vert \sum_{i=1}^{N}f^{i}(x,t) \Biggr\vert ^{ \frac{p(x,t)}{p(x,t)-1}}a(x,t)^{-\frac{1}{p(x,t)-1}}\,dx< c,\quad i=1,2, \ldots , N, t\in [0,T], \end{aligned}$$
(4.1)
$$\begin{aligned}& \operatorname{div}\vec{f}(x,t)\geq 0, \end{aligned}$$
(4.2)

then

$$ u(x,t)=v(x,t),\quad (x,t)\in Q_{T}. $$
(4.3)

Proof

For a small positive constant \(\delta >0\), denoting \(D_{\delta }=\{x \in \varOmega : w=u-v>\delta \}\), we suppose that the measure \(\mu (D _{\delta })>0\). Let

$$ F_{\lambda }(\xi )= \textstyle\begin{cases} \frac{1}{1-\beta }\lambda ^{\beta -1}-\frac{1}{1-\beta }\xi ^{\beta -1}, & \text{if } \xi >{\lambda }, \\ 0, & \text{if } \xi \leq \lambda , \end{cases} $$
(4.4)

where \(\delta >2\lambda >0\), \(1>\beta >0\).

Now, by a process of limit, we can choose \(F_{\lambda }(w)=F_{\lambda }(u-v)\) and integrate it over \(Q_{t}\), \(0\leq t< T\), accordingly,

$$ \begin{aligned}[b] 0&= \int _{0}^{t} \int _{\varOmega } \bigl[w_{t}F_{\lambda }(w)+a(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)-2}\nabla u - \vert \nabla v \vert ^{p(x,t)-2}\nabla v\bigr) \nabla F _{\lambda }(w) \bigr] \,dx \,dt \\ &\quad {}+\sum_{i=1}^{N} \int _{0}^{t} \int _{\varOmega }\bigl(u^{q}-v^{q}\bigr) \bigl[f^{i}(x,t) \bigl(F _{\lambda }(u-v)\bigr)_{x_{i}} \bigr] \,dx \,dt \\ &\quad {}+\sum_{i=1}^{N} \int _{0}^{t} \int _{\varOmega }\bigl(u^{q}-v^{q}\bigr) \bigl[f^{i} _{x_{i}}(x,t)F_{\lambda }(u-v) \bigr] \,dx \,dt. \end{aligned} $$
(4.5)

In the first place,

$$ \begin{gathered}[b] \int _{0}^{t} \int _{\varOmega } a(x,t) \bigl( \vert \nabla u \vert ^{p(x,t)-2}\nabla u - \vert \nabla v \vert ^{p(x,t)-2}\nabla v \bigr) \cdot \nabla (u-v)F'_{\lambda }(u-v) \,dx \,dt \\ \quad \geq \int _{0}^{t} \int _{\varOmega } a(x,t) (u-v)^{2-\beta }2^{-p(x,t)} \vert \nabla w \vert ^{p(x,t)} \,dx \,dt \\ \quad \geq 2^{-p_{+}} \int _{0}^{t} \int _{\varOmega } a(x,t)w^{2-\beta } \vert \nabla w \vert ^{p(x,t)} \,dx \,dt \\ \quad \geq 0. \end{gathered} $$
(4.6)

By that \(q>0\) and condition (4.1),

$$ \int _{\varOmega }\sum_{i=1}^{N}f^{i}(x,t)^{\frac{p(x,t)}{p(x,t)-1}}a^{- \frac{1}{p(x,t)-1}}(x) \,dx< c,\quad i=1,2,\ldots , N, t\in [0,T], $$

using the last formula of (3.3) and the dominated convergence theorem, we have

$$ \begin{gathered}[b] \Biggl\vert \sum _{i=1}^{N} \int _{0}^{t} \int _{\varOmega }f^{i}(x,t) \bigl(u^{q}-v^{q} \bigr)F _{\lambda }'(u-v) (u-v)_{x_{i}} \,dx \,dt \Biggr\vert \\ \quad = \Biggl\vert \int _{0}^{t} \int _{\varOmega }\sum_{i=1}^{N}f^{i}(x,t)a(x,t)^{- \frac{1}{p(x,t)}} \bigl(u^{q}-v^{q}\bigr)F_{\lambda }'(u-v)a^{\frac{1}{p(x,t)}}(u-v)_{x _{i}} \,dx \,dt \Biggr\vert \\ \quad \leq c \int _{0}^{t} \int _{\varOmega }w^{2-\beta } \Biggl( \Biggl\vert \sum _{i=1} ^{N}f^{i}(x,t) \Biggr\vert a(x,t)^{-\frac{1}{p(x,t)}}\bigl(u^{q}-v^{q}\bigr) \Biggr) ^{\frac{p(x,t)}{p(x,t)-1}} \\ \qquad {}+2^{-p_{+}-1} \int _{0}^{t} \int _{\varOmega }a(x,t)w^{2-\beta } \vert \nabla w \vert ^{p(x,t)} \,dx \,dt \\ \quad \leq c+\frac{2^{-p_{+}}}{2} \int _{0}^{T} \int _{\varOmega }a(x,t)w^{2- \beta } \vert \nabla w \vert ^{p(x,t)} \,dx \,dt. \end{gathered} $$
(4.7)

Since \(\operatorname{div}\vec{f}(x)\geq 0\),

$$ \begin{gathered}[b] \int _{0}^{t} \int _{\varOmega } \sum_{i=1}^{N}f_{x_{i}}^{i}(x,t) \bigl(u^{q}-v ^{q}\bigr)F_{\lambda }(u-v) \,dx \,dt \\ \quad = \int _{0}^{t} \int _{\varOmega }\operatorname{div}\vec{f}(x,t) \bigl(u^{q}-v^{q}\bigr)\frac{1}{ \beta -1}(u-v)^{\beta -1} \,dx \,dt \\ \qquad {}- \int _{0}^{t} \int _{\varOmega }\operatorname{div}\vec{f}(x,t) \bigl(u^{q}-v^{q}\bigr)\frac{1}{ \beta -1}\lambda ^{\beta -1} \,dx \,dt \\ \quad \geq - \int _{0}^{t} \int _{D_{\lambda }}\operatorname{div}\vec{f}(x,t) \bigl(u^{q}-v ^{q}\bigr)\frac{1}{\beta -1}\lambda ^{\beta -1} \,dx \,dt \\ \quad \geq -c_{1}, \end{gathered} $$
(4.8)

where \(c_{1}\) is independent of λ.

Moreover, let \(t_{0}=\inf \{\tau \in (0,t]: w>\lambda \}\). Then

$$ \begin{aligned}[b] \int _{0}^{t} \int _{D_{\lambda }} w_{t}F_{\lambda }(w) \,dx \,dt&= \int _{D_{\lambda }} \biggl( \int _{0}^{t_{0}}w_{t}F_{\lambda }(w) \,dt+ \int _{t_{0}}^{t}w_{t}F_{\lambda }(w) \,dt \biggr)\,dx \\ &\geq \int _{D_{\lambda }} \int _{\lambda }^{w(x,t)}F_{\lambda }(s)\,ds \,dx \\ &\geq \int _{D_{\lambda }}(w-2\lambda )F_{\lambda }(2\lambda )\,dx \geq ( \delta -2\lambda )F_{\lambda }(2\lambda )\mu (D_{\lambda }). \end{aligned} $$
(4.9)

Thus, we have

$$ (\delta -2\lambda )\frac{1-2^{\beta -1}}{1-\beta }\leq c_{2}, $$

where \(c_{1}\) is independent of λ. Letting \(\lambda \rightarrow 0\), we get the contradiction. □

Proof of Theorem 1.4

By conditions (1.13) and (1.14), only if we choose \(\phi (x,t)=a(x,t)\), we know conditions (3.5) (3.6) in Theorem 3.3 are true, the conclusion follows easily. □

Asymptotic behavior of weak solutions

In what follows, \(p(x,t)=p(x)\), \(p_{+}=\max_{x\in \overline{\varOmega }}p(x)\), \(p_{-}= \min_{x\in \overline{\varOmega }}p(x)\), \(q(x)=\frac{p(x)}{p(x)-1}\).

Lemma 5.1

Let \(p,s\in C_{+}(\overline{\varOmega })\) and \(a(x)\) satisfy

(w1):

\(a \in L_{\mathrm{loc}}^{1}(\varOmega )\) and \({a^{-\frac{1}{{p(x) - 1}}}} \in L_{\mathrm{loc}}^{1}(\varOmega )\);

(w2):

\({a^{ - s(x)}} \in {L^{1}}(\varOmega )\) with \(s(x) \in (\frac{N}{ {p(x)}},\infty ) \cap [\frac{1}{{p(x) - 1}},\infty )\). Then we have the following compact embedding:

$$ {W^{1,p(x)}}(a,\varOmega )\hookrightarrow \hookrightarrow {L^{r(x)}}( \varOmega ) $$

provided that \(r\in C_{+}(\overline{\varOmega })\) and \(1\leq r(x)< p_{s} ^{\ast }(x)\) for all \(x\in \varOmega \). Here,

$$ p_{s}(x)=\frac{p(x)s(x)}{1+s(x)}, $$

and

$$ p_{s}^{\ast }(x)=\textstyle\begin{cases} \frac{p(x)s(x)N}{(s(x)+1)N-p(x)s(x)}, & \textit{if } p_{s}(x)< N, \\ +\infty , & \textit{if } p_{s}(x)\geq N. \end{cases} $$

Lemma 5.2

Let \(p\in C_{+}(\overline{\varOmega })\). If (w1) and (w2) hold, then the estimate

$$ { \Vert u \Vert _{{L^{p(x)}}(\varOmega )}} \leqslant C{ \Vert {\nabla u} \Vert _{{L^{p(x)}}(a,\varOmega )}} $$

holds for every \(u\in C_{0}^{\infty }(\varOmega )\) with a positive constant C independent of u.

These two lemmas and the definitions about the weighted variable exponent Sobolev space \(W^{1,p(x)}(a,\varOmega )\) can be found in [32].

Theorem 5.3

Suppose that \(a(x,t)=a(x)\) satisfying (1.6), (w1) and (w2), \(p(x,t)=p(x)\geq p_{-}>1\) and \(f_{i}(x,t)=f_{i}(x)\) satisfies (4.1) and (4.2), \(2<\frac{Np_{+}}{N-p_{+}}\). If there is \(0<\alpha <\frac{p _{-}}{p_{+}(p_{+}-p_{-})}\) such that

$$\begin{aligned}& \int _{\varOmega }a(x) \vert \nabla u \vert ^{p(x)} \,dx\geq M(u_{0})\geq 1,\qquad \int _{ \varOmega }a(x) \vert \nabla u \vert ^{p(x)} \,dx\leq ct^{\alpha },\quad t>1, \end{aligned}$$
(5.1)
$$\begin{aligned}& \lim_{T\rightarrow \infty } \int _{0}^{T}\frac{1}{1+t^{\alpha \beta }}\,dx= \infty , \end{aligned}$$
(5.2)
$$\begin{aligned}& \Biggl( \int _{\varOmega } \Biggl\vert \sum_{i=1}^{N}f^{i}(x)a(x)^{- \frac{1}{p(x)}}u^{q} \Biggr\vert ^{\frac{p(x)}{p(x)-1}}\,dx \Biggr)^{\frac{1}{q ^{+}}}< 1, \end{aligned}$$
(5.3)

where \(\beta ^{-1}=\frac{p_{-}}{p_{+}(p_{+}-p_{-})}\), then

$$ \lim_{T\rightarrow \infty } \bigl\Vert u(x,T) \bigr\Vert _{L^{2}(\varOmega )}=0. $$
(5.4)

Proof

Let \(G(u)=\frac{1}{2}\int _{\varOmega }|u|^{2}\,dx\). Then it is well known that G is a convex functional on \(L^{2}(\varOmega )\). For any \(t\in (0,T)\) and \(h>0\), δ represents Găteaux differential, i.e.,

$$ \frac{\delta G(u)}{\delta u}=u. $$

By the convexity of G, we have

$$ G\bigl(u(t+h)\bigr)-G\bigl(u(t)\bigr)\geq \int _{\varOmega }\bigl[u(x,t+h)-u(x,t)\bigr]u(x,t)\,dx. $$
(5.5)

For any \(t_{1}, t_{2}\in [0,T]\), \(t_{1}< t_{2}\),

$$ \begin{gathered}[b] \int _{t_{1}}^{t_{2}}G\bigl(u(t+h)\bigr)\,dt- \int _{t_{1}}^{t_{2}}G\bigl(u(t)\bigr)\,dt \\ \quad = \int _{t_{1}+h}^{t_{2}+h}G\bigl(u(t)\bigr)\,dt- \int _{t_{1}}^{t_{2}}G\bigl(u(t)\bigr)\,dt \\ \quad = \int _{t_{2}}^{t_{2}+h}G\bigl(u(t+h)\bigr)\,dt- \int _{t_{1}}^{t_{+}h}G\bigl(u(t)\bigr)\,dt \\ \quad \geq \int _{t_{1}}^{t_{2}} \int _{\varOmega }\bigl[u(x,t+h)-u(x,t)\bigr]u(x,t)\,dx. \end{gathered} $$
(5.6)

Dividing both sides of (5.6) by h, we let \(h\rightarrow 0\). Then

$$ G\bigl(u(t_{2})\bigr)-G\bigl(u(t_{1})\bigr)\geq \int _{t_{1}}^{t_{2}} \int _{\varOmega }\frac{ \partial u}{\partial t}u(x,t) \,dx \,dt. $$

In a similar way, we have

$$ G\bigl(u(t)\bigr)-G\bigl(u(t-h)\bigr)\leq \int _{\varOmega }\bigl[u(x,t)-u(x,t-h)\bigr]u(x,t)\,dx, $$
(5.7)

accordingly,

$$ G\bigl(u(t_{2})\bigr)-G\bigl(u(t_{1})\bigr)\leq \int _{t_{1}}^{t_{2}} \int _{\varOmega }\frac{ \partial u}{\partial t}u(x,t) \,dx \,dt. $$

Then

$$ G\bigl(u(t_{2})\bigr)-G\bigl(u(t_{1}) \bigr)= \int _{t_{1}}^{t_{2}} \int _{\varOmega }\frac{ \partial u}{\partial t}u(x,t) \,dx \,dt. $$
(5.8)

In particular, by the definition of weak solution, we have

$$ G\bigl(u(t)\bigr)-G\bigl(u(0)\bigr)=- \int _{0}^{t} \int _{\varOmega }a(x) \vert \nabla u \vert ^{p(x)} \,dx \,dt+ \int _{0}^{t} \int _{\varOmega }\vec{f}(x)\cdot \nabla u^{q} \,dx \,dt. $$
(5.9)

By Theorem 4.1, the solution of equation (1.1) with initial (1.2) is unique, then we can regard it as the limit

$$ u(x,t)=\lim_{\varepsilon \rightarrow 0}u_{\varepsilon }(x,t), $$

where \(u_{\varepsilon }\) is the solution of the initial boundary value problem (2.1)–(2.3). Thus,

$$ u_{\varepsilon }\in L^{p_{+}}\bigl(0,T; W_{0}^{1,p_{+}}(\varOmega )\bigr),\qquad u_{ \varepsilon t}\in L^{{p'}_{+}}\bigl(0,T;W^{-1,{p'}_{+}}(\varOmega )\bigr) $$
(5.10)

by that \(2<\frac{Np_{+}}{N-p_{+}}\), we have

$$ W_{0}^{1,p_{+}}(\varOmega )\hookrightarrow \text{(compact) } L^{2}( \varOmega )\hookrightarrow W^{-1,{p'}_{+}}( \varOmega ). $$
(5.11)

Thus, \(u(x,t)\in C(0,T; L^{2}(\varOmega ))\).

Let \(G(t)=\frac{1}{2}\int _{\varOmega }|u(x,t)|^{2}\,dx\). Then \(G(t)\) is continuous in \([0,T)\), and by (5.1)(5.3), we have

$$ \begin{aligned}[b] G'(t)&=- \int _{\varOmega }a(x) \vert \nabla u \vert ^{p(x)} \,dx-\sum_{i=1}^{N} \int _{ \varOmega } \bigl[f^{i}(x)u^{q}u_{x_{i}}+f_{x_{i}}^{i}(x,t)u^{q+1} \bigr]\,dx \\ &\leq - \int _{\varOmega }a(x) \vert \nabla u \vert ^{p(x)} \,dx-\sum_{i=1}^{N} \int _{ \varOmega }f^{i}(x)u^{q}u_{x_{i}} \,dx \\ &\leq - \int _{\varOmega }a(x) \vert \nabla u \vert ^{p(x)} \,dx \\ &\quad {}+ \Biggl( \int _{\varOmega } \Biggl\vert \sum_{i=1}^{N}f^{i}(x)a(x)^{- \frac{1}{p(x)}}u^{q} \Biggr\vert ^{\frac{p(x)}{p(x)-1}}\,dx \Biggr)^{\frac{1}{q _{1}}} \biggl( \int _{\varOmega }a(x) \vert \nabla u \vert ^{p(x)} \,dx \biggr)^{\frac{1}{p _{1}}} \\ &\leq 0. \end{aligned} $$
(5.12)

Here, \(p_{1}=p_{+}\) or \(p_{-}\) according to \(\int _{\varOmega }a(x)| \nabla u|^{p(x)}\,dx\geq 1\) or \(\int _{\varOmega }a(x)|\nabla u|^{p(x)}\,dx<1\), \(q_{1}\) has a similar meaning.

We choose \(s(x)=2\) in Lemma 5.1, \(p_{s}(x)=\frac{2p(x)}{3}\). If \(p_{s}(x)< N\) and \(3N-2p(x)< Np(x)\), then \(2< p_{s}^{*}(x)= \frac{2Np(x)}{3N-2p(x)}\). If \(p_{s}(x)\geq N\), \(p_{s}^{*}=\infty \), \(2< p_{s}^{*}\) is naturally. By Lemma 5.1,

$$ {W^{1,p(x)}}(a,\varOmega )\hookrightarrow \hookrightarrow {L^{2}}(\varOmega ). $$

Accordingly, by Lemma 5.2 and assumption (5.1), we are able to show that

$$ \begin{aligned}[b] \int _{\varOmega } \vert u \vert ^{2}\,dx&\leq c \Vert u \Vert _{W^{1,p(x)}(a,\varOmega )}\leq c { \Vert {\nabla u} \Vert _{{L^{p(x)}}(w,\varOmega )}} \\ &\leq ct^{\frac{2\alpha (p_{+}-p_{-})}{p_{-}}} \bigl\vert G'(t) \bigr\vert ^{\frac{2}{p _{+}}},\quad t\geq 1. \end{aligned} $$
(5.13)

By (5.13), we can extrapolate that

$$ \int _{\varOmega } \bigl\vert u(x,T) \bigr\vert ^{2}\,dx\leq \frac{1}{(c_{1}\int _{0}^{T}(1+t^{ \alpha \beta })^{-1}+c_{2})^{\delta }},\quad \delta =\frac{2}{p_{+}-2}, c _{1}>0, c_{2}>0. $$

This accomplishes the proof of the theorem. □

Conclusion

The initial boundary value problem of an electrorheological fluid equation with orientated convection term is considered. The diffusion coefficient \(a(x,t)\), the variable exponent \(p(x,t)\), and the oriented convection coefficient \(b_{i}(x,t)\) are all dependent on time variable t. If, for any t, \(a(x,t)=0\), \(x\in \partial \varOmega \), then the stability of weak solutions may be true without boundary value condition. This conclusion generalizes our previous works [28, 29]. The essential improvement lies in that only if \(a(x,t)|_{x\in \partial \varOmega }=0\) for any t, the uniqueness of weak solution is always true, no other conditions are required. Just by this important result, we can study the large time behavior of weak solutions without the boundary value condition. To the best knowledge, this is the first paper to study the large time behavior for an initial boundary value problem but independent of the boundary value condition.

References

  1. 1.

    Ruzicka, M.: Electrorheological Fluids: Modeling and Mathematical Theory. Lecture Notes in Math., vol. 1748. Springer, Berlin (2000)

  2. 2.

    Antontsev, S., Shmarev, S.: Anisotropic parabolic equations with variable nonlinearity. Publ. Mat. 53, 355–399 (2009)

  3. 3.

    Gao, W., Guo, B.: Existence and localization of weak solutions of nonlinear parabolic equations with variable exponent of nonlinearity. Ann. Mat. Pura Appl. 191, 551–562 (2012)

  4. 4.

    Guo, B., Gao, W.: Study of weak solutions for parabolic equations with nonstandard growth conditions. J. Math. Anal. Appl. 374(2), 374–384 (2011)

  5. 5.

    Bennouna, J., El hamdaoui, B., Mekkour, M., Redwane, H.: Nonlinear parabolic inequalities in Lebesgue–Sobolev spaces with variable exponent. Ric. Mat. 65(1), 93–125 (2016)

  6. 6.

    Bendahmane, M., Wittbold, P., Zimmermann, A.: Renormalized solutions for a nonlinear parabolic equation with variable exponents and \(L^{1}\) data. J. Differ. Equ. 249(6), 1483–1515 (2010)

  7. 7.

    Wu, Z., Zhao, J., Yun, J., Li, F.: Nonlinear Diffusion Equations. World Scientific, Singapore (2001)

  8. 8.

    DiBenedetto, E.: Degenerate Parabolic Equations. Springer, New York (1993)

  9. 9.

    Zhao, J.: Existence and nonexistence of solutions for \({u_{t}} =\operatorname{div}( { \vert {\nabla u} \vert ^{p - 2}}\nabla u) + f(\nabla u,u,x,t)\). J. Math. Anal. Appl. 172, 130–146 (1993)

  10. 10.

    Lee, K., Petrosyan, A., Vazquez, J.: Large time geometric properties of solutions of the evolution p-Laplacian equation. J. Differ. Equ. 229, 389–411 (2006)

  11. 11.

    Nakao, M.: \(L^{p}\) estimates of solutions of some nonlinear degenerate diffusion equation. J. Math. Soc. Jpn. 37, 41–63 (1985)

  12. 12.

    Ragnedda, F., Vernier, P., Vespri, V.: Large time behavior of solutions to a class of non-autonomous degenerate parabolic equations. Math. Ann. 348, 779–795 (2010)

  13. 13.

    DiBenedetto, E., Gianazza, U., Vespri, V.: Harnack type estimates and Hölder continuity for nonnegative solutions to certain sub-critically singular parabolic partial differential equations. Manuscr. Math. 131, 231–245 (2010)

  14. 14.

    Ragusa, M.A., Tachikawa, A.: On continuity of minimizers for certain quadratic growth functionals. J. Math. Soc. Jpn. 57(3), 691–700 (2005)

  15. 15.

    Zhan, H.: On the well-posedness problem of the electrorheological fluid equations. Bound. Value Probl. 2018, 104 (2018)

  16. 16.

    Zhan, H.: The stability of the anisotropic parabolic equation with the variable exponent. Bound. Value Probl. 2017, 134 (2017)

  17. 17.

    Giacomoni, J., Rădulescu, V.D., Warnault, G.: Quasilinear parabolic problem with variable exponent: qualitative analysis and stabilization. Commun. Contemp. Math. 20(8), 1750065 (2018)

  18. 18.

    Ho, K., Sim, I.: A priori bounds and existence for solutions of weighted elliptic equations with a convection term. Adv. Nonlinear Anal. 6(4), 427–445 (2017)

  19. 19.

    Rădulescu, V.D.: Nonlinear elliptic equations with variable exponent: old and new. Nonlinear Anal. 121, 336–369 (2015)

  20. 20.

    Rădulescu, V.D., Repovš, D.D.: Partial differential equations with variable exponents. Variational methods and qualitative analysis. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2015)

  21. 21.

    Repovš, D.D.: Stationary waves of Schrödinger-type equations with variable exponent. Anal. Appl. 13(6), 645–661 (2015)

  22. 22.

    Zhan, H.: The weak solutions of an evolutionary \(p(x)\)-Laplacian equation are controlled by the initial value. Comput. Math. Appl. 76, 2272–2285 (2018)

  23. 23.

    Zhan, H., Feng, Z.: Solutions of evolutionary \(p(x)\)-Laplacian equation based on the weighted variable exponent space. Z. Angew. Math. Phys. 68, 134 (2017)

  24. 24.

    Yin, J., Wang, C.: Properties of the boundary flux of a singular diffusion process. Chin. Ann. Math. 25B(2), 175–182 (2004)

  25. 25.

    Yin, J., Wang, C.: Evolutionary weighted p-Laplacian with boundary degeneracy. J. Differ. Equ. 237, 421–445 (2007)

  26. 26.

    Zhan, H.: The uniqueness of the solution to the diffusion equation with a damping term. Appl. Anal. 98(7), 1333–1346 (2019)

  27. 27.

    Ye, H., Yin, J.: Propagation profile for a non-Newtonian polytropic filtration equation with orientated convection. J. Math. Anal. Appl. 421, 1225–1237 (2015)

  28. 28.

    Zhan, H., Feng, Z.: Stability of the solutions of a convection-diffusion equation. Nonlinear Anal. 182, 193–208 (2019)

  29. 29.

    Zhan, H.: Stability of weak solutions of a non-Newtonian polytropic filtration equation. Electron. J. Differ. Equ. 2018, 190, 1–18 (2018)

  30. 30.

    Bin, G., Gao, W.: Existence and asymptotic behavior of solutions for nonlinear parabolic equations with variable exponent of nonlinearity. Acta Math. Sci. 32B(3), 1053–1062 (2012)

  31. 31.

    Zhikov, V.V.: On the density of smooth functions in Sobolev–Orlicz spaces. Zap. Nauč. Semin. POMI 310, 67–81 (2004) translation in J. Math. Sci. (N.S.), 132, 285–294 (2006)

  32. 32.

    Ho, K., Sim, I.: On degenerate \(p(x)\)-Laplacian equations involving critical growth with two parameters. Nonlinear Anal. 132, 95–114 (2016)

  33. 33.

    Aubin, J.P.: Un théorème de compacité. C. R. Acad. Sci. 256, 5042–5044 (1963)

Download references

Acknowledgements

The author would like to thank to reviewers for their good comments on my paper.

Availability of data and materials

No applicable.

Funding

The paper is supported by the Natural Science Foundation of Fujian province (2019J01858), and SF of Xiamen University of Technology, China.

Author information

The author read and approved the final manuscript.

Correspondence to Huashui Zhan.

Ethics declarations

Competing interests

The author declares that he has no competing interests.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

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.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

MSC

  • 35K65
  • 35K55
  • 35R92

Keywords

  • The electrorheological fluid equation
  • Orientated convection term
  • Partial boundary value condition
  • Stability
  • Large time behavior