Let \(\boldsymbol {\varphi }(t)\) be given and let (2.5) hold for some \(\delta>0\). The aim of this section is to prove Theorem 2.1.
3.1 An approximation problem
We will construct an approximation problem of (1.1a)-(1.3). We first construct some approximation functions.
For an arbitrary \(\varepsilon >0\), choose \(\boldsymbol {\varphi }_{\varepsilon }=(\varphi ^{1}_{\varepsilon }(t),\ldots, \varphi ^{n}_{\varepsilon }(t))\in \mathbf {C}^{2}([0, T])\) such that \(\boldsymbol {\varphi }_{\varepsilon }(t)\to \boldsymbol {\varphi }(t)\) in \(\mathbf {W}^{1}_{2}(0, T)\) as \(\varepsilon \to0\). Then, for small enough ε,
$$\begin{aligned} \|\boldsymbol {\varphi }_{\varepsilon }\|_{\mathbf {W}^{1}_{2}(0, T)}\leq\|\boldsymbol {\varphi }\|_{\mathbf {W}^{1}_{2}(0, T)} +1. \end{aligned}$$
(3.1)
Let \(s_{\varepsilon }=s_{\varepsilon }(\theta)\) be a smooth function with values between 0 and 1 such that \(|\mathrm{d}s_{\varepsilon }(\theta)/\mathrm{d}\theta|\leq C/\varepsilon \) for all \(\theta\in\mathbb{R}\), \(s_{\varepsilon }(\theta)=1\) for \(\theta\leq0\) and \(s_{\varepsilon }(\theta)=0\) for \(\theta\geq \varepsilon \), and let
$$\begin{aligned} z_{\varepsilon ,k}(x,t):=\left \{ \begin{array}{@{}l} \prod_{\tau=1}^{n}s_{\varepsilon }(x-\varphi ^{\tau}_{\varepsilon }(t))\quad ((x,t)\in Q_{T}), k=1,\\ \prod_{\vartheta=1}^{n} [1-s_{\varepsilon }(x-\varphi ^{\vartheta}_{\varepsilon }(t))]\quad ((x,t)\in Q_{T}), k=n+1,\\ \prod_{\tau=k}^{n}s_{\varepsilon }(x-\varphi ^{\tau}_{\varepsilon }(t)) \prod_{\vartheta=1}^{k-1} [1-s_{\varepsilon }(x-\varphi ^{\vartheta}_{\varepsilon }(t)) ] \quad((x,t)\in Q_{T}), k=2,\ldots,n. \end{array} \right . \end{aligned}$$
Define
$$\begin{aligned}& A^{l}_{\varepsilon }(x,t):=\left \{ \begin{array}{@{}l} \hat {a}^{l} s_{\varepsilon }(x-\varphi ^{l}_{\varepsilon }(t))+\tilde {a}^{l}[1-s_{\varepsilon }(x-\varphi ^{l}_{\varepsilon }(t))] \quad((x,t)\in Q_{T}), l=1,\ldots,n,\\ a^{l} \quad((x,t)\in Q_{T}), l=n+1,\ldots,N, \end{array} \right . \end{aligned}$$
(3.2)
$$\begin{aligned}& B^{l}_{\varepsilon }\bigl(x,t,u^{l} \bigr):= \left \{ \begin{array}{@{}l} \hat {b}^{l}(u^{l}) s_{\varepsilon }(x-\varphi ^{l}_{\varepsilon }(t))+\tilde {b}^{l}(u^{l})[1-s_{\varepsilon }(x-\varphi ^{l}_{\varepsilon }(t))]\\ \quad ((x,t)\in Q_{T}), l=1,\ldots,n,\\ b^{l}(u^{l})\quad ((x,t)\in Q_{T}), l=n+1,\ldots,N, \end{array} \right . \end{aligned}$$
(3.3)
and
$$ G_{\varepsilon }^{l}(x,t,\mathbf {u}):=\sum_{k=1}^{n+1} g^{l}_{k}(\mathbf {u}) z_{\varepsilon ,k}(x,t) \quad \bigl((x,t)\in Q_{T} \bigr), l=1,\ldots,N. $$
Then according to hypothesis (H)(ii) and (iii), it follows that the vector function \(\mathbf{G}_{\varepsilon }(\cdot, \mathbf {u})=(G^{1}_{\varepsilon }(\cdot, \mathbf {u}),\ldots,G^{N}_{\varepsilon }(\cdot, \mathbf {u}))\) is mixed quasimonotone in with index vector \(([\mathbf {u}]_{\rho^{1}};\ldots;[\mathbf {u}]_{\rho^{N}})\), and
(3.4)
$$\begin{aligned}& \nu\leq A^{l}_{\varepsilon }(x,t)\leq C,\qquad B^{l}_{\varepsilon }\bigl(x,t,u^{l} \bigr)|\leq\mu \bigl(\bigl|u^{l}\bigr| \bigr), \end{aligned}$$
(3.5)
(3.6)
where \(\nu:=\min_{i=1,\ldots,n, j=n+1,\ldots,N}\{\hat {a}^{i},\tilde {a}^{i},a^{j}\}\). The definition of function \(s_{\varepsilon }(\theta)\) implies that
$$\begin{aligned}& A^{l}_{\varepsilon }(x,t)= \left \{ \begin{array}{@{}l@{\quad}l} \hat {a}^{l} &(x-\varphi ^{l}_{\varepsilon }(t)\leq0),\\ \tilde {a}^{l} &(x-\varphi ^{l}_{\varepsilon }(t)\geq \varepsilon ), \end{array} \right . \end{aligned}$$
(3.7)
$$\begin{aligned}& B^{l}_{\varepsilon }\bigl(x,t,u^{l} \bigr)= \left \{ \begin{array}{@{}l@{\quad}l} \hat {b}^{l}(u^{l}) &(x-\varphi ^{l}_{\varepsilon }(t)\leq0),\\ \tilde {b}^{l}(u^{l}) &(x-\varphi ^{l}_{\varepsilon }(t)\geq \varepsilon ), \end{array} \right .\quad l=1,\ldots,n. \end{aligned}$$
(3.8)
In addition, it is obvious from (2.5) that
$$ \bigcup_{\tau=1}^{k}\bar{Q}_{\tau,T}- \Gamma ^{k}_{T}= \bigl\{ (x,t):x-\varphi ^{k}(t)<0 \bigr\} \cap\bar{Q}_{T}, $$
(3.9)
where \(\bar{Q}_{T}\), \(\bar{Q}_{\tau,T}\) are the closure of \({Q}_{T}\) and \({Q}_{\tau,T}\), respectively. Thus by (3.9) and the definition of functions \(z_{\varepsilon ,k}(x,t)\), an argument similar to the one used in [16, Lemma 3.2] shows that
$$ \sum_{k=1}^{n+1} z_{\varepsilon ,k}(x,t)= 1 \quad \bigl((x,t)\in\bar{Q}_{T} \bigr), $$
and
$$\begin{aligned} G_{\varepsilon }^{l}(x,t,\mathbf {u}) = \left \{ \begin{array}{@{}l} g^{l}_{1}(\mathbf {u}) \quad\mbox{if }x-\varphi ^{1}_{\varepsilon }(t)\leq0,\\ g^{l}_{n+1}(\mathbf {u}) \quad\mbox{if } x-\varphi ^{n}_{\varepsilon }(t)\geq \varepsilon ,\\ g^{l}_{k}(\mathbf {u}) \quad\mbox{if } x-\varphi ^{k-1}_{\varepsilon }(t)\geq \varepsilon \mbox{ and } x-\varphi ^{k}_{\varepsilon }(t)\leq0\\ \quad\mbox{for some } k\in\{2,\ldots,n\},\\ g^{l}_{k-1}(\mathbf {u}) s_{\varepsilon }(x-\varphi ^{k-1}_{\varepsilon }(t)) +g^{l}_{k}(\mathbf {u})[1-s_{\varepsilon }(x-\varphi ^{k-1}_{\varepsilon }(t))]\\ \quad\mbox{if } 0< x-\varphi ^{k-1}_{\varepsilon }(t)<\varepsilon \mbox{ for some } k\in\{2,\ldots,n\}. \end{array} \right . \end{aligned}$$
(3.10)
We next construct the approximation functions of \(\psi^{l}(x,t)\). Let \(\omega(|x|)\) be a sufficiently smooth nonnegative function such that \(\omega(|x|)=0\) for \(|x|\geq1\) and \(\int_{|x|\leq 1}\omega(x)\,\mathrm{d}x=1\), and let \(\lambda =\lambda (x)\) be a sufficiently smooth nonnegative function taking values in \([0,1]\) such that \(\lambda (x)=0\) for \(\delta\leq x\leq d-\delta\), \(\lambda (x)=1\) for \(x\leq\delta/2\) or \(x\geq d-\delta/2\), and \(\lambda _{x}(x)\leq C/\delta\) for all \(x\in \mathbb{R}\). Define
$$ \psi_{\varepsilon }^{l}=\psi_{\varepsilon }^{l}(x,t):=\int _{|x-y|\leq \varepsilon }\omega\bigl(|x-y|\bigr) \bigl(1-\lambda (y) \bigr)\psi^{l}(y,t) \,\mathrm{d}y+\lambda (x)\psi^{l}(x,t),\quad l=1,\ldots,N. $$
Then hypothesis (H)(i) and [17, Chapter II] imply that
$$ \left \{ \begin{array}{@{}l} \psi_{\varepsilon }^{l}(x,t)\in C^{\alpha _{0}}(\bar{Q}_{T})\cap W^{1,1}_{2}(Q_{T}), \qquad\psi_{\varepsilon }^{l}(x,0)\in C^{2+\alpha _{0}}([0,d]),\\ \psi_{\varepsilon }^{l}(0,t),\psi_{\varepsilon }^{l}(d,t)\in C^{2+\alpha _{0}}([0,T]),\quad l=1,\ldots,N, \end{array} \right . $$
(3.11)
and
$$\begin{aligned}& \psi_{\varepsilon }^{l}(x,t)\to\psi^{l}(x,t) \quad\mbox{in } C^{\alpha _{0}}(\bar{Q}_{T}) \mbox{ and in }W^{1,1}_{2}(Q_{T}), \\& \psi_{\varepsilon }^{l}(x,0)\to\psi^{l}(x,0) \quad\mbox{in } H^{1}(0,d), \end{aligned}$$
and (2.1) and (2.3) imply that
$$\begin{aligned}& m^{l}\leq\psi^{l}_{\varepsilon }(x,t)\leq M^{l} \quad \bigl((x,t)\in S_{T}\cup \bigl\{ \Omega \times\{0\} \bigr\} \bigr), \end{aligned}$$
(3.12)
$$\begin{aligned}& \bigl\| \psi^{l}_{\varepsilon }(x,t)\bigr\| _{C^{\alpha _{0}}(\bar{Q}_{T})}\leq \mu_{1}+1, \end{aligned}$$
(3.13)
$$\begin{aligned}& \begin{aligned}[b] &\bigl\| \psi^{l}_{\varepsilon }(x,t)\bigr\| _{W^{1,1}_{2}(Q_{T})}+\bigl\| \psi^{l}_{\varepsilon }(x,0)\bigr\| _{H^{1}(0, d)}+\bigl\| \psi^{l}_{\varepsilon x}(x,0) \bigr\| _{C([0, \delta_{0}])} \\ &\quad{} +\bigl\| \psi^{l}_{\varepsilon x}(x,0)\bigr\| _{C([d-\delta_{0}, d])}\leq C(1/ \delta),\quad l=1,\ldots,N. \end{aligned} \end{aligned}$$
(3.14)
In addition, for \(\varepsilon <\delta/8\),
$$ \psi^{l}_{\varepsilon }(x,t)=\psi^{l}(x,t) \quad \bigl((x,t)\in \bigl\{ [0,\delta/4]\cup[d-\delta/4] \bigr\} \times [0,T] \bigr). $$
(3.15)
Employing the above approximation functions, we consider the following approximation problem:
$$ \left \{ \begin{array}{@{}l} u^{l}_{t}-(A^{l}_{\varepsilon }(x,t)u^{l}_{x})_{x}=B^{l}_{\varepsilon }(x,t,u^{l})u^{l}_{x}+G^{l}_{\varepsilon }(x,t,\mathbf {u})\quad ((x, t)\in Q_{T}),\\ u^{l}=\psi^{l}_{\varepsilon }(x,t)\quad ((x,t)\in S_{T}\cup([0,d]\times \{0\} ) ), l=1,\ldots,N. \end{array} \right . $$
(3.16)
Lemma 3.1
Problem (3.16) has a unique classical solution
\(\mathbf {u}_{\varepsilon }\)
in
, and the following estimates hold:
$$\begin{aligned}& \bigl\| u^{l}_{\varepsilon }\bigr\| _{C^{\alpha _{2}, \alpha _{2}/2}(\bar{Q}_{T})}\leq C\quad \bigl( \alpha _{2}\in(0, 1) \bigr), \end{aligned}$$
(3.17)
$$\begin{aligned}& \bigl\| u^{l}_{\varepsilon }\bigr\| _{V^{1,0}_{2}(Q_{T})}\leq C(1/ \delta), \end{aligned}$$
(3.18)
$$\begin{aligned}& \bigl|u^{l}_{x}(0,t)\bigr|+\bigl|u^{l}_{x}(d,t)\bigr| \leq C(1/\delta) \quad \bigl(t\in[0,T] \bigr), \end{aligned}$$
(3.19)
$$\begin{aligned}& \max_{[0,T]}\bigl\| u^{l}_{\varepsilon x} \bigr\| _{L^{2}(0,d)}+\bigl\| u^{l}_{\varepsilon t}\bigr\| _{L^{2}(Q_{T})}+\bigl\| \bigl(A^{l}_{\varepsilon }(x,t) u^{l}_{\varepsilon x} \bigr)_{x}\bigr\| _{L^{2}(Q_{T})} \leq C\bigl(1/\delta,\|\boldsymbol {\varphi }_{t}\|_{\mathbf {L}^{2}(0,T)}\bigr), \end{aligned}$$
(3.20)
$$\begin{aligned}& \bigl\| u^{l}_{ \varepsilon x}\bigr\| _{L^{\infty, 4}(Q_{T})} \leq C\bigl(1/\delta,\| \boldsymbol {\varphi }_{t}\|_{\mathbf {L}^{2}(0,T)}\bigr),\quad l=1,\ldots,N, \end{aligned}$$
(3.21)
where constants
\(\alpha _{2}\)
and
C
are independent of
ε.
Proof
In [15], by using the method of upper and lower solutions, together with the associated monotone iterations and various estimates, we investigated the existence and uniqueness of the global piecewise classical solutions of the quasilinear parabolic system with discontinuous coefficients and continuous delays under various conditions including mixed quasimonotone property of reaction functions. The same problem was also discussed for the system with continuous coefficients without time-delay.
It is obvious that problem (3.16) is the special case of [15, problem (1.1)] without discontinuous coefficients and time delays. Hypothesis (H)(ii) shows that \(\tilde{\mathbf {u}}=\mathbf{M}\), \(\hat{\mathbf {u}}=\mathbf{m}\) are the coupled weak upper and lower solutions of (3.16) in the sense of [15, Definition 2.2]. By (3.4)-(3.6) and (3.11)-(3.14), we conclude from [15, Theorem 4.1] that problem (3.16) has a unique classical solution \(\mathbf {u}_{\varepsilon }\) in . Furthermore, using (3.1), (3.5), (3.6) and (3.12)-(3.14), the proof similar to that of [16, Lemma 3.3] shows that estimates (3.17) and (3.18) hold.
To prove (3.19), we first fix \(l\in\{1,\ldots,n\}\). In view of (3.7), (3.8), (3.10) and (3.15), we find that \(u^{l}_{\varepsilon }\) is the solution of the following problems for single equation:
$$ \left \{ \begin{array}{@{}l} u^{l}_{\varepsilon t}-(\hat {a}^{l} u^{l}_{\varepsilon x})_{x}=\hat {b}^{l}(u^{l}_{\varepsilon })u^{l}_{\varepsilon x}+g^{l}_{1}(u^{l}_{\varepsilon },[\mathbf {u}_{\varepsilon }]_{\rho^{l}},[\mathbf {u}_{\varepsilon }]_{\omega^{l}})\quad ((x,t)\in(0,\delta/4)\times(0,T]),\\ u^{l}_{\varepsilon }=\psi^{l}(x,t) \quad(x=0, 0\leq t\leq T \mbox{ or } 0\leq x\leq\delta/4, t=0), \end{array} \right . $$
(3.22)
and
$$ \left \{ \begin{array}{@{}l} u^{l}_{\varepsilon t}-(\tilde {a}^{l} u^{l}_{\varepsilon x})_{x}=\tilde {b}^{l}(u^{l}_{\varepsilon })u^{l}_{\varepsilon x}+g^{l}_{n+1}(u^{l}_{\varepsilon },[\mathbf {u}_{\varepsilon }]_{\rho^{l}},[\mathbf {u}_{\varepsilon }]_{\omega^{l}})\\ \quad((x,t)\in(d-\delta/4,d)\times(0,T]),\\ u^{l}_{\varepsilon }=\psi^{l}(x,t) \quad(x=d, 0\leq t\leq T \mbox{ or } d-\delta/4\leq x\leq d, t=0). \end{array} \right . $$
(3.23)
By (3.22), (3.23), (3.12) and (3.14), the proof similar to that of [17, Chapter VI, Lemma 3.1] gives (3.19) for \(l\in\{1,\ldots,n\}\). The similar argument shows that (3.19) holds for \(l\in\{n+1,\ldots,N\}\).
We next prove (3.20). For any fixed \(l\in\{1,\ldots,n\}\), let
$$ y=x-\varphi ^{l}_{\varepsilon }(t),\qquad t'=t\quad (\mbox{denoted by } t \mbox{ still}), $$
and let
$$ q(y, t)=u^{l}_{\varepsilon }\bigl(y+\varphi ^{l}_{\varepsilon }(t),t \bigr). $$
By a direct computation we have
$$ \left \{ \begin{array}{lc}q_{y}(y, t)=u^{l}_{\varepsilon x}(x,t), \qquad q_{t}(y,t)=q_{y}(y,t)\varphi ^{l}_{\varepsilon t}+u^{l}_{\varepsilon t},\\ q_{y}(d-\varphi ^{l}_{\varepsilon }(t),t)=u^{l}_{\varepsilon x}(d,t),\qquad q_{y}(-\varphi ^{l}_{\varepsilon }(t),t)=u^{l}_{\varepsilon x}(0,t),\\ q_{t}(d-\varphi ^{l}_{\varepsilon }(t),t)=u^{l}_{\varepsilon x}(d,t)\varphi ^{l}_{\varepsilon t}+\psi^{l}_{\varepsilon t}(d,t),\\ q_{t}(-\varphi ^{l}_{\varepsilon }(t),t)=u^{l}_{\varepsilon x}(0,t)\varphi ^{l}_{\varepsilon t}+\psi^{l}_{\varepsilon t}(0,t). \end{array} \right . $$
(3.24)
Then (3.16), (3.2), (3.3) and (3.10) imply that the function \(q=q(y,t)\) satisfies
$$ \left \{ \begin{array}{@{}l} q_{t}-\{[\hat {a}^{l} s_{\varepsilon }(y)+\tilde {a}^{l}(1-s_{\varepsilon }(y))]q_{y}\}_{y}-q_{y}\varphi ^{l}_{\varepsilon t}\\ \quad=B^{l}_{\varepsilon }(y+\varphi ^{l}_{\varepsilon }(t),t,q)q_{y}+G^{l}_{\varepsilon }(y+\varphi ^{l}_{\varepsilon }(t),t,\mathbf {u}_{\varepsilon }(y+\varphi ^{l}_{\varepsilon }(t),t))\quad ((y,t)\in E^{l}_{T}),\\ q(-\varphi ^{l}_{\varepsilon }(t), t)=\psi^{l}_{\varepsilon }(0,t),\qquad q(d-\varphi ^{l}_{\varepsilon }(t), t)=\psi ^{l}_{\varepsilon }(d,t) \quad (t\in[0,T]),\\ q(y, 0)=\psi^{l}_{\varepsilon }(y+\varphi ^{l}_{\varepsilon }(0),0)\quad (y\in[-\varphi ^{l}_{\varepsilon }(0), d-\varphi ^{l}_{\varepsilon }(0)]), \end{array} \right . $$
(3.25)
where \(E^{l}_{T}:=\{(y, t):-\varphi ^{l}_{\varepsilon }(t)< y< d-\varphi ^{l}_{\varepsilon }(t), 0<t\leq T\}\).
A double integration by parts gives
$$\begin{aligned} &{-}\iint_{E^{l}_{\tau}} \bigl\{ \bigl[\hat {a}^{l} s_{\varepsilon }(y)+ \tilde {a}^{l} \bigl(1-s_{\varepsilon }(y) \bigr) \bigr]q_{y} \bigr\} _{y} q_{t}\,\mathrm{d}y\, \mathrm{d}t \\ &\quad =\frac{1}{2}\int_{-\varphi ^{l}_{\varepsilon }(t)}^{d-\varphi ^{l}_{\varepsilon }(t)} \bigl[ \hat {a}^{l} s_{\varepsilon }(y)+\tilde {a}^{l} \bigl(1-s_{\varepsilon }(y) \bigr) \bigr](q_{y})^{2}\,\mathrm{d}y \Big|_{t=0}^{t=\tau} \\ &\qquad{} +\int_{0}^{\tau} \bigl[\hat {a}^{l} s_{\varepsilon }(y)+\tilde {a}^{l} \bigl(1-s_{\varepsilon }(y) \bigr) \bigr] \biggl\{ -q_{y}q_{t}+\frac{1}{2}\varphi ^{l}_{\varepsilon t}(q_{y})^{2} \biggr\} \Big|_{y=-\varphi ^{l}_{\varepsilon }}^{y=d-\varphi ^{l}_{\varepsilon }}\,\mathrm{d}t. \end{aligned}$$
(3.26)
Thus multiplying the equation in (3.25) by \(q_{t}\), integrating it on \(E^{l}_{\tau}\) and using (3.5), (3.6), (3.24) and (3.26), we find that
$$\begin{aligned} &\frac{1}{2}\int_{-\varphi ^{l}_{\varepsilon }(\tau)}^{d-\varphi ^{l}_{\varepsilon }(\tau)} \bigl[ \hat {a}^{l} s_{\varepsilon }(y)+\tilde {a}^{l} \bigl(1-s_{\varepsilon }(y) \bigr) \bigr] \bigl(q_{y}(y,\tau) \bigr)^{2}\,\mathrm{d}y + \iint_{E^{l}_{\tau}}(q_{t})^{2}\,\mathrm{d}y\,\mathrm{d}t \\ &\quad \leq C\int_{0}^{d} \bigl( \psi^{l}_{x}(x,0) \bigr)^{2}\,\mathrm{d}x+C \iint_{E^{l}_{\tau}} \bigl(\bigl| \varphi ^{l}_{\varepsilon t}\bigr|+1 \bigr) \bigl(|q_{y}|+1\bigr)|q_{t}| \,\mathrm{d}x\,\mathrm{d}t \\ &\qquad{} +C\int_{0}^{\tau}\bigl\{ \bigl|u^{l}_{\varepsilon x}(0,t)\bigr| \bigl[\bigl|u^{l}_{\varepsilon x}(0,t) \varphi ^{l}_{\varepsilon t}\bigr|+\bigl| \psi^{l}_{\varepsilon t}\bigr| \bigr]+\bigl|u^{l}_{\varepsilon x}(d,t)\bigr| \bigl[\bigl|u^{l}_{\varepsilon x}(d,t) \varphi ^{l}_{\varepsilon t}\bigr|+\bigl| \psi^{l}_{\varepsilon t}\bigr| \bigr] \\ &\qquad{} +\bigl|\varphi ^{l}_{\varepsilon t}\bigr| \bigl[ \bigl(u^{l}_{\varepsilon x}(0,t) \bigr)^{2}+ \bigl(u^{l}_{\varepsilon x}(d,t) \bigr)^{2} \bigr] \bigr\} \,\mathrm{d}t. \end{aligned}$$
Furthermore, by (3.12)-(3.14), (3.19) and Cauchy’s inequality, we deduce that for any \(\sigma>0\),
$$\begin{aligned} &\frac{1}{2}\nu\int_{-\varphi ^{l}_{\varepsilon }(\tau)}^{d-\varphi ^{l}_{\varepsilon }(\tau)} \bigl(q_{y}(y,\tau) \bigr)^{2}\,\mathrm{d}y + \iint_{E^{l}_{\tau}}(q_{t})^{2} \,\mathrm{d}y\,\mathrm{d}t \\ &\quad\leq\sigma\iint_{E^{l}_{\tau}}(q_{t})^{2}\, \mathrm{d}y\, \mathrm{d} t +C(\sigma)\iint_{E^{l}_{\tau}} \bigl[1+(q_{y})^{2} \bigr] \bigl[1+ \bigl(\varphi ^{l}_{\varepsilon t} \bigr)^{2} \bigr] \,\mathrm{d}y\,\mathrm{d}t+C\bigl(1/\delta,\|\boldsymbol {\varphi }_{t} \|_{\mathbf {L}^{2}(0,T)}\bigr). \end{aligned}$$
Choosing \(\sigma=\nu/4\), we get
$$\begin{aligned} &\int_{-\varphi ^{l}_{\varepsilon }(\tau)}^{d-\varphi ^{l}_{\varepsilon }(\tau )} \bigl(q_{y}(y,\tau) \bigr)^{2}\,\mathrm{d}y + \iint_{E^{l}_{\tau}}(q_{t})^{2} \,\mathrm{d}y\,\mathrm{d}t \\ &\quad \leq\int_{0}^{\tau}\biggl\{ \bigl[1+ \bigl( \varphi ^{l}_{\varepsilon t} \bigr)^{2} \bigr]\int _{-\varphi ^{l}_{\varepsilon }(t)}^{d-\varphi ^{l}_{\varepsilon }(t)} \bigl(q_{y}(y,t) \bigr)^{2}\,\mathrm{d}y \biggr\} \,\mathrm{d}t+C\bigl(1/\delta,\| \boldsymbol {\varphi }_{t} \|_{\mathbf {L}^{2}(0,T)}\bigr). \end{aligned}$$
(3.27)
Consequently,
$$\begin{aligned} & \bigl[1+ \bigl(\varphi ^{l}_{\varepsilon t}(\tau) \bigr)^{2} \bigr]\int_{-\varphi ^{l}_{\varepsilon }(\tau)}^{d-\varphi ^{l}_{\varepsilon }(\tau )} \bigl(q_{y}(y,\tau) \bigr)^{2}\,\mathrm{d}y \\ &\quad \leq \bigl[1+ \bigl(\varphi ^{l}_{\varepsilon t}(\tau) \bigr)^{2} \bigr] \int_{0}^{\tau}\biggl\{ \bigl[1+ \bigl(\varphi ^{l}_{\varepsilon t} \bigr)^{2} \bigr]\int _{-\varphi ^{l}_{\varepsilon }(t)}^{d-\varphi ^{l}_{\varepsilon }(t)} \bigl(q_{y}(y,t) \bigr)^{2}\,\mathrm{d}y \biggr\} \,\mathrm{d}t \\ &\qquad{} + \bigl[1+ \bigl(\varphi ^{l}_{\varepsilon t}(\tau) \bigr)^{2} \bigr]C\bigl(1/\delta,\|\boldsymbol {\varphi }_{t}\|_{\mathbf {L}^{2}(0,T)}\bigr). \end{aligned}$$
This, together with Gronwall’s inequality (see [17, Chapter II, Lemma 5.5]), implies that
$$\begin{aligned} \iint_{E^{l}_{\tau}} \bigl[1+ \bigl( \varphi ^{l}_{\varepsilon t} \bigr)^{2} \bigr] \bigl[1+(q_{y})^{2} \bigr]\, \mathrm{d}y \,\mathrm{d}t\leq C\bigl(1/\delta,\|\boldsymbol {\varphi }_{t}\|_{\mathbf {L}^{2}(0,T)}\bigr). \end{aligned}$$
Hence we deduce from (3.25), (3.27), (3.5) and (3.6) that
$$\begin{aligned} &\max_{ 0\leq t\leq T}\int_{-\varphi ^{l}_{\varepsilon }(t)}^{d-\varphi ^{l}_{\varepsilon }(t)} (q_{y})^{2}\,\mathrm{d}y +\iint_{E^{l}_{T}} \bigl\{ (q_{t})^{2}+ \bigl[ \bigl( \bigl(\hat {a}^{l} s_{\varepsilon }(y)+\tilde {a}^{l} \bigl(1-s_{\varepsilon }(y) \bigr) \bigr)q_{y} \bigr)_{y} \bigr]^{2} \bigr\} \, \mathrm{d}y\,\mathrm{d} t \\ &\quad \leq C\bigl(1/\delta,\|\boldsymbol {\varphi }_{t}\|_{\mathbf {L}^{2}(0,T)}\bigr), \end{aligned}$$
which, together with (3.24), yields (3.20) for \(l\in \{1,\ldots,n\}\).
For any fixed \(l\in\{n+1,\ldots,N\}\), we consider the equality
$$ \iint_{Q_{\tau}} \bigl\{ u^{l}_{\varepsilon t}- \bigl(A^{l}_{\varepsilon }(x,t)u^{l}_{\varepsilon x} \bigr)_{x}-B^{l}_{\varepsilon }\bigl(x,t,u^{l}_{\varepsilon }\bigr)u^{l}_{\varepsilon x}-G^{l}_{\varepsilon }(x,t, \mathbf {u}_{\varepsilon }) \bigr\} u^{l}_{\varepsilon t}\,\mathrm{d}x\, \mathrm{d}t=0. $$
A similar argument gives (3.20) for \(l\in\{n+1,\ldots,N\}\). Therefore, (3.20) holds for all \(l\in\{1,\ldots,N\}\).
It remains to prove (3.21). For each \(l\in\{1,\ldots,N\}\), since \(A^{l}_{\varepsilon }(x,t)u^{l}_{\varepsilon x}\) is in \(V_{2}(Q_{T})\), then (3.20) and [17, Chapter 2, formula (3.8)] show that
$$\begin{aligned} \nu\bigl\| u^{l}_{\varepsilon x}\bigr\| _{L^{\infty,4}(Q_{T})} \leq& \bigl\| A^{l}_{\varepsilon }(x,t)u^{l}_{\varepsilon x} \bigr\| _{L^{\infty,4}(Q_{T})} \\ \leq& \bigl\| \bigl(A^{l}_{\varepsilon }(x,t)u^{l}_{\varepsilon x} \bigr)_{x}\bigr\| _{L^{2}(Q_{T})}+\sup_{0\leq t\leq T} \bigl\| A^{l}_{\varepsilon }(x,t)u^{l}_{\varepsilon x} \bigr\| _{L^{2}(0,d)} \\ \leq&C\bigl(1/\delta,\|\boldsymbol {\varphi }_{t}\|_{\mathbf {L}^{2}(0,T)}\bigr). \end{aligned}$$
Thus (3.21) holds. □
Lemma 3.2
Let
\(\mathbf {u}_{\varepsilon _{1}}(x,t)\), \(\mathbf {u}_{\varepsilon _{2}}(x,t)\)
be the solutions in
for problem (3.16) corresponding to
\(\boldsymbol {\varphi }_{\varepsilon _{1}}(t)\)
and
\(\boldsymbol {\varphi }_{\varepsilon _{2}}(t)\), respectively. Then
$$\begin{aligned} \|\mathbf {u}_{\varepsilon _{1}}-\mathbf {u}_{\varepsilon _{2}}\|_{\mathcal{V}^{1,0}_{2}(\mathcal {Q}_{T})}^{2} \leq C\bigl(1/\delta,\|\boldsymbol {\varphi }_{t}\|_{\mathbf {L}^{2}(0,T)}\bigr)I_{3,0}, \end{aligned}$$
(3.28)
where
\(I_{3,0}:=\sum_{l=1}^{N}\|\psi^{l}_{\varepsilon _{1}}(x,0)-\psi^{l}_{\varepsilon _{2}}(x,0)\| _{H^{1}(0,d)}^{2} +\|\boldsymbol {\varphi }_{\varepsilon _{1}}-\boldsymbol {\varphi }_{\varepsilon _{2}}\|_{\mathbf {L}^{2}(0,T)}+\varepsilon _{1}+\varepsilon _{2}\).
Proof
Let \(\mathbf {w}=(w^{1},\ldots,w^{N})=\mathbf {u}_{\varepsilon _{1}}-\mathbf {u}_{\varepsilon _{2}}\), and let \(l\in\{1,\ldots,N\}\) be fixed. We see from (3.16) that
$$ \left \{ \begin{array}{@{}l} w^{l}_{t}= \{ [A^{l}_{\varepsilon _{1}}(x,t)-A^{l}_{\varepsilon _{2}}(x,t) ]u^{l}_{\varepsilon _{1} x} \}_{x} + \{A^{l}_{\varepsilon _{2}}(x,t)w^{l}_{x} \}_{x}\\ \hphantom{w^{l}_{t}=}{}+ [B^{l}_{\varepsilon _{1}}(x,t,u^{l}_{\varepsilon _{1}})u^{l}_{\varepsilon _{1} x} -B^{l}_{\varepsilon _{2}}(x,t,u^{l}_{\varepsilon _{2}})u^{l}_{\varepsilon _{2} x} ]\\ \hphantom{w^{l}_{t}=}{}+ [G_{\varepsilon _{1}}^{l}(x,t,\mathbf {u}_{\varepsilon _{1}}) -G_{\varepsilon _{2}}^{l}(x,t,\mathbf {u}_{\varepsilon _{2}}) ]\quad ((x, t)\in Q_{T}), \\ w^{l}=\psi^{l}_{\varepsilon _{1}}(x,t)-\psi^{l}_{\varepsilon _{2}}(x,t)\quad ((x,t)\in S_{T}\cup( [0,d ]\times\{0\} ) ). \end{array} \right . $$
(3.29)
In view of (3.15), we find \(w^{l}(0,t)=w^{l}(d,t)=0\). Multiplying the equation in (3.29) by \(w^{l}\) and integrating by parts on \(Q_{\tau}\), we deduce that, for any \(\tau\in[0,T]\),
$$\begin{aligned} &\frac{1}{2}\int_{0}^{d} \bigl(w^{l} \bigr)^{2}\,\mathrm{d}x \Big|_{t=0}^{t=\tau} + \iint_{Q_{\tau}}A^{l}_{\varepsilon _{2}}(x,t) \bigl(w^{l}_{x} \bigr)^{2}\,\mathrm{d}x\,\mathrm{d}t \\ &\quad =\iint_{Q_{\tau}}- \bigl[A^{l}_{\varepsilon _{1}}(x,t)-A^{l}_{\varepsilon _{2}}(x,t) \bigr]u^{l}_{\varepsilon _{1} x}w^{l}_{x}\,\mathrm{d}x \,\mathrm{d}t \\ &\qquad{} +\iint_{Q_{\tau}} \bigl[B^{l}_{\varepsilon _{1}} \bigl(x,t,u^{l}_{\varepsilon _{1}} \bigr)u^{l}_{\varepsilon _{1} x} -B^{l}_{\varepsilon _{2}} \bigl(x,t,u^{l}_{\varepsilon _{2}} \bigr)u^{l}_{\varepsilon _{2} x} \bigr]w^{l}\,\mathrm{d}x\, \mathrm{d}t \\ &\qquad{} +\iint_{Q_{\tau}} \bigl[G_{\varepsilon _{1}}^{l}(x,t, \mathbf {u}_{\varepsilon _{1}}) -G_{\varepsilon _{2}}^{l}(x,t,\mathbf {u}_{\varepsilon _{2}}) \bigr]w^{l}\,\mathrm{d}x\,\mathrm{d}t \\ &\quad =:I^{l}_{3,1}+I^{l}_{3,2}+I^{l}_{3,3}. \end{aligned}$$
(3.30)
Let us estimate \(I^{l}_{3,1}\), \(I^{l}_{3,2}\) and \(I^{l}_{3,3}\). Since (3.2), (3.3) and (3.10) imply that
$$\begin{aligned}& A^{l}_{\varepsilon _{1}}(x,t)-A^{l}_{\varepsilon _{2}}(x,t) = \left \{ \begin{array}{@{}l} {[}s_{\varepsilon _{1}}(x-\varphi ^{l}_{\varepsilon _{1}}(t))-s_{\varepsilon _{2}}(x-\varphi ^{l}_{\varepsilon _{2}}(t)) ](\hat {a}^{l}-\tilde {a}^{l}),\quad l=1,\ldots,n,\\ 0,\quad l=n+1,\ldots,N, \end{array} \right . \\& \begin{aligned}[b] &\bigl|B^{l}_{\varepsilon _{1}} \bigl(x,t,u^{l}_{\varepsilon _{1}} \bigr)u^{l}_{\varepsilon _{1} x} -B^{l}_{\varepsilon _{2}} \bigl(x,t,u^{l}_{\varepsilon _{2}} \bigr)u^{l}_{\varepsilon _{2} x} \bigr| \\ &\quad =\left \{ \begin{array}{@{}l} |[s_{\varepsilon _{1}}(x-\varphi ^{l}_{\varepsilon _{1}}(t))-s_{\varepsilon _{2}}(x-\varphi ^{l}_{\varepsilon _{2}}(t)) ] [\hat {b}^{l}(u^{l}_{\varepsilon _{1}})-\tilde {b}^{l}(u^{l}_{\varepsilon _{1}}) ]u^{l}_{\varepsilon _{1} x}\\ \quad{}+s_{\varepsilon _{2}}(x-\varphi ^{l}_{\varepsilon _{2}}(t)) [\hat {b}^{l}(u^{l}_{\varepsilon _{1}})u^{l}_{\varepsilon _{1} x}-\hat {b}^{l}(u^{l}_{\varepsilon _{2}})u^{l}_{\varepsilon _{2} x} ]\\ \quad{}+[1-s_{\varepsilon _{2}}(x-\varphi ^{l}_{\varepsilon _{2}}(t)) ] [\tilde {b}^{l}(u^{l}_{\varepsilon _{1}})u^{l}_{\varepsilon _{1} x}-\tilde {b}^{l}(u^{l}_{\varepsilon _{2}})u^{l}_{\varepsilon _{2} x} ] |,\quad l=1,\ldots,n,\\ |{[}b^{l}(u^{l}_{\varepsilon _{1}})-b^{l}(u^{l}_{\varepsilon _{2}}) ]u^{l}_{\varepsilon _{1} x}+b^{l}(u^{l}_{\varepsilon _{2}})(u^{l}_{\varepsilon _{1} x}-u^{l}_{\varepsilon _{2} x}) |,\quad l=n+1,\ldots,N \end{array} \right . \\ &\quad \leq C\sum_{l=1}^{n} \bigl| \bigl[s_{\varepsilon _{1}} \bigl(x-\varphi ^{l}_{\varepsilon _{1}}(t) \bigr)-s_{\varepsilon _{2}} \bigl(x-\varphi ^{l}_{\varepsilon _{2}}(t) \bigr) \bigr] \bigr|\bigl| u^{l}_{\varepsilon _{1} x}\bigr|+C\sum_{l=1}^{N} \bigl[\bigl|w^{l}\bigr|\bigl|u^{l}_{\varepsilon _{1}x}\bigr|+\bigl|w^{l}_{x}\bigr| \bigr], \end{aligned} \end{aligned}$$
and
$$\begin{aligned} &G_{\varepsilon _{1}}^{l}(x,t,\mathbf {u}_{\varepsilon _{1}}) -G_{\varepsilon _{2}}^{l}(x,t, \mathbf {u}_{\varepsilon _{2}}) \\ &\quad =\left \{ \begin{array}{@{}l} g^{l}_{1}(\mathbf {u}_{\varepsilon _{1}})-g^{l}_{1}(\mathbf {u}_{\varepsilon _{2}}) \quad\mbox{if } x\leq\min(\varphi ^{1}_{\varepsilon _{1}}(t),\varphi ^{1}_{\varepsilon _{2}}(t)),\\ g^{l}_{n+1}(\mathbf {u}_{\varepsilon _{1}})-g^{l}_{n+1}(\mathbf {u}_{\varepsilon _{2}}) \quad\mbox{if } x\geq\max(\varphi ^{n}_{\varepsilon _{1}}(t)+\varepsilon _{1},\varphi ^{n}_{\varepsilon _{2}}(t)+\varepsilon _{2}),\\ g^{l}_{k}(\mathbf {u}_{\varepsilon _{1}})-g^{l}_{k}(\mathbf {u}_{\varepsilon _{2}}) \quad\mbox{if } x\leq\min(\varphi ^{k}_{\varepsilon _{1}}(t),\varphi ^{k}_{\varepsilon _{2}}(t)) \mbox{ and}\\ \quad x\geq\max(\varphi ^{k-1}_{\varepsilon _{1}}(t)+\varepsilon _{1},\varphi ^{k-1}_{\varepsilon _{2}}(t)+\varepsilon _{2}) \mbox{ for } k\in\{2,\ldots,n\},\\ G_{\varepsilon _{1}}^{l}(x,t,\mathbf {u}_{\varepsilon _{1}}) -G_{\varepsilon _{2}}^{l}(x,t,\mathbf {u}_{\varepsilon _{2}}) \\ \quad\mbox{if } \min(\varphi ^{k}_{\varepsilon _{1}}(t),\varphi ^{k}_{\varepsilon _{2}}(t))< x< \max(\varphi ^{k}_{\varepsilon _{1}}(t)+\varepsilon _{1},\varphi ^{k}_{\varepsilon _{2}}(t)+\varepsilon _{2}) \quad\mbox{for } k\in\{1,\ldots,n\}, \end{array} \right . \end{aligned}$$
then it follows from Cauchy’s inequality that, for any \(\sigma>0\),
$$\begin{aligned} &I^{l}_{3,1}\leq\sigma\iint_{Q_{\tau}}| \mathbf {w}_{x}|^{2}\,\mathrm{d}x\,\mathrm{d}t+C( \sigma)I_{3,4}, \end{aligned}$$
(3.31)
$$\begin{aligned} &I^{l}_{3,2}\leq\sigma\iint_{Q_{\tau}}| \mathbf {w}_{x}|^{2}\,\mathrm{d}x\,\mathrm{d}t+C(\sigma) \iint_{Q_{\tau}}| \mathbf {w}|^{2}\,\mathrm{d}x\,\mathrm{d}t+CI_{3,4}+CI_{3,5} \end{aligned}$$
(3.32)
and
$$\begin{aligned} I^{l}_{3,3} \leq& \sum _{k=1}^{n+1} \iint_{Q_{\tau}}\bigl|g^{l}_{k}( \mathbf {u}_{\varepsilon _{1}}) -g^{l}_{k}(\mathbf {u}_{\varepsilon _{2}})\bigr|\bigl|w^{l}\bigr| \, \mathrm{d}x\,\mathrm{d}t \\ &{}+\sum_{k=1}^{n}\int _{0}^{\tau}\int_{\min\{\varphi ^{k}_{\varepsilon _{1}},\varphi ^{k}_{\varepsilon _{2}}\}}^{\max\{\varphi ^{k}_{\varepsilon _{1}}+\varepsilon _{1},\varphi ^{k}_{\varepsilon _{2}}+\varepsilon _{2}\}}\bigl|G_{\varepsilon _{1}}^{l}(x,t, \mathbf {u}_{\varepsilon _{1}}) -G_{\varepsilon _{2}}^{l}(x,t,\mathbf {u}_{\varepsilon _{2}})\bigr|\bigl|w^{l}\bigr| \,\mathrm{d}x\,\mathrm{d}t \\ \leq& C\iint_{Q_{\tau}}|\mathbf {w}|^{2}\,\mathrm{d}x\,\mathrm{d}t+C\sum _{k=1}^{n}\bigl\| \bigl|\varphi ^{k}_{\varepsilon _{1}}- \varphi ^{k}_{\varepsilon _{2}}\bigr|+\varepsilon _{1}+\varepsilon _{2} \bigr\| _{L^{2}(0,\tau)}, \end{aligned}$$
(3.33)
where
$$\begin{aligned}& I_{3,4}:=\sum_{l=1}^{n} \iint_{Q_{\tau}} \Biggl[1+\sum_{j=1}^{2} \bigl(u^{l}_{\varepsilon _{j}x} \bigr)^{2} \Biggr] \bigl[s_{\varepsilon _{1}} \bigl(x-\varphi ^{l}_{\varepsilon _{1}}(t) \bigr)-s_{\varepsilon _{2}} \bigl(x-\varphi ^{l}_{\varepsilon _{2}}(t) \bigr) \bigr]^{2} \,\mathrm{d}x\,\mathrm{d}t, \\& I_{3,5}:=\sum_{l=1}^{N} \iint_{Q_{\tau}} \Biggl[1+\sum_{j=1}^{2} \bigl(u^{l}_{\varepsilon _{j}x} \bigr)^{2} \Biggr] \bigl(w^{l} \bigr)^{2}\,\mathrm{d}x\,\mathrm{d}t. \end{aligned}$$
According to the definition of function \(s_{\varepsilon }(\theta)\), we see that \(s_{\varepsilon _{1}} (x-\varphi ^{l}_{\varepsilon _{1}}(t))-s_{\varepsilon _{2}}(x-\varphi ^{l}_{\varepsilon _{2}}(t))=0\) if \(x<\min\{\varphi ^{l}_{\varepsilon _{1}}(t), \varphi ^{l}_{\varepsilon _{2}}(t)\}\) or \(x>\max\{\varphi ^{l}_{\varepsilon _{1}}(t)+\varepsilon _{1},\varphi ^{l}_{\varepsilon _{2}}(t)+\varepsilon _{2}\}\). Thus by (3.21) we have
$$\begin{aligned} I_{3,4} \leq&\sum_{l=1}^{n} \Biggl\{ \int_{0}^{\tau}\Biggl(1+\sum _{l,j=1}^{2}\bigl\| u^{l}_{\varepsilon _{j}x}\bigr\| ^{2}_{L^{\infty}(0,d)} \Biggr)^{2}\,\mathrm{d}t \Biggr\} ^{1/2} \\ &{} \times \biggl\{ \int_{0}^{\tau}\biggl[\int _{0}^{d} \bigl(s_{\varepsilon _{1}} \bigl(x- \varphi ^{l}_{\varepsilon _{1}}(t) \bigr)-s_{\varepsilon _{2}} \bigl(x- \varphi ^{l}_{\varepsilon _{2}}(t) \bigr) \bigr)^{2}\,\mathrm{d}x \biggr]^{2}\,\mathrm{d}t \biggr\} ^{1/2} \\ \leq& C\bigl(1/\delta,\|\boldsymbol {\varphi }_{t}\|_{\mathbf {L}^{2}(0,T)}\bigr)\sum _{l=1}^{n} \biggl\{ \int_{0}^{\tau}\biggl[\int_{\min\{\varphi ^{l}_{\varepsilon _{1}},\varphi ^{l}_{\varepsilon _{2}}\}}^{\max\{\varphi ^{l}_{\varepsilon _{1}}+\varepsilon _{1},\varphi ^{l}_{\varepsilon _{2}}+\varepsilon _{2}\}}1\,\mathrm{d}x \biggr]^{2} \,\mathrm{d}t \biggr\} ^{1/2} \\ \leq&C\bigl(1/\delta,\|\boldsymbol {\varphi }_{t}\|_{\mathbf {L}^{2}(0,T)}\bigr)\sum _{l=1}^{n} \bigl\| \bigl|\varphi ^{l}_{\varepsilon _{1}}- \varphi ^{l}_{\varepsilon _{2}}\bigr|+\varepsilon _{1}+\varepsilon _{2} \bigr\| _{L^{2}(0,\tau)} \end{aligned}$$
(3.34)
and
$$\begin{aligned} I_{3,5}\leq\int_{0}^{\tau}\Biggl[ \Biggl(1+\sum_{j=1}^{2}\|\mathbf {u}_{\varepsilon _{j}x} \|^{2}_{\mathbf {L}^{\infty}(0,d)} \Biggr)\int_{0}^{d} |\mathbf {w}|^{2}\,\mathrm{d}x \Biggr]\,\mathrm{d}t. \end{aligned}$$
(3.35)
Summing equality (3.30) with respect to l from \(l=1\) to \(l=N\), using (3.31)-(3.35) and Minkowski’s inequality, and choosing \(\sigma=\nu/8\), we then conclude that
$$\begin{aligned} &\int_{0}^{d} |\mathbf {w}|^{2}(x,\tau)\, \mathrm{d}x + \int_{0}^{\tau}\int_{0}^{d}| \mathbf {w}_{x}|^{2}\,\mathrm{d}x\,\mathrm{d}t \\ &\quad \leq C\bigl(1/\delta,\|\boldsymbol {\varphi }_{t}\|_{\mathbf {L}^{2}(0,T)}\bigr) \Biggl\{ \int _{0}^{\tau}\Biggl[ \Biggl(1+\sum _{j=1}^{2}\|\mathbf {u}_{\varepsilon _{j}x}\|^{2}_{\mathbf {L}^{\infty}(0,d)} \Biggr)\int_{0}^{d} |\mathbf {w}|^{2}\, \mathrm{d}x \Biggr]\,\mathrm{d}t+I_{3,0} \Biggr\} . \end{aligned}$$
This, together with Gronwall’s inequality, yields
$$\begin{aligned} &\int_{0}^{\tau}\Biggl[ \Biggl(1+\sum _{j=1}^{2}\|\mathbf {u}_{\varepsilon _{j}x}\|^{2}_{\mathbf {L}^{\infty}(0,d)} \Biggr)\int_{0}^{d} |\mathbf {w}|^{2}\, \mathrm{d}x \Biggr]\,\mathrm{d}t \\ &\quad \leq C\bigl(1/\delta,\|\boldsymbol {\varphi }_{t}\|_{\mathbf {L}^{2}(0,T)}\bigr)I_{3,0} \exp \Biggl\{ C\int_{0}^{T} \Biggl(1+\sum _{j=1}^{2}\|\mathbf {u}_{\varepsilon _{j}x}\|^{2}_{\mathbf {L}^{\infty}(0,d)} \Biggr)\,\mathrm{d}t \Biggr\} \\ &\qquad{} \times\int_{0}^{T} \Biggl(1+\sum _{j=1}^{2}\|\mathbf {u}_{\varepsilon _{j}x} \|^{2}_{\mathbf {L}^{\infty}(0,d)} \Biggr)\,\mathrm{d}t. \end{aligned}$$
Combining the two inequalities above and (3.21) leads us to estimate (3.28). □
3.2 The solutions of the diffraction problem
Proof of Theorem 2.1
We divide the proof into three steps.
Step 1. We prove the global existence of the solutions. Let us discuss the behavior of the solution \(\mathbf {u}_{\varepsilon }\) associated with \(\boldsymbol {\varphi }_{\varepsilon }(t)\) by Theorem 2.1 as \(\varepsilon \to0\).
We first see from (3.16) that for any \(\tau\in[0,T]\) and any vector function ,
$$\begin{aligned} &\int_{0}^{d} u^{l}_{\varepsilon }\eta^{l}\,\mathrm{d}x \Big|_{0}^{\tau}+ \iint_{Q_{\tau}} \bigl\{ -u^{l}_{\varepsilon }\eta^{l}_{t}+A^{l}_{\varepsilon }(x,t)u^{l}_{\varepsilon x}\eta^{l}_{x}- \bigl[B^{l}_{\varepsilon }\bigl(x,t, u^{l}_{\varepsilon }\bigr)u^{l}_{\varepsilon x}+G^{l}_{\varepsilon }(x,t, \mathbf {u}_{\varepsilon }) \bigr]\eta^{l} \bigr\} \,\mathrm{d}x\,\mathrm{d}t \\ &\quad =0,\quad l=1,\ldots,N. \end{aligned}$$
(3.36)
Furthermore, according to estimates (3.17), (3.20), (3.28) and the Arzela-Ascoli theorem, we conclude that there exists a subsequence (we retain the same notation for it) \(\{\mathbf {u}_{\varepsilon }\}\) such that
$$\begin{aligned}& \mathbf {u}_{\varepsilon }\to \mathbf {u}\quad\mbox{in }\mathcal{C}(\bar{Q}_{T}), \qquad \mathbf {u}_{\varepsilon t}\rightharpoonup \mathbf {u}_{t} \quad\mbox{weakly in } \mathcal {L}^{2}(Q_{T}),\\& \mathbf {u}_{\varepsilon }\to \mathbf {u}\quad \mbox{in } \mathcal{V}_{2}^{1, 0}(\mathcal{Q}_{T}). \end{aligned}$$
Thus u is in \(\mathcal{C}^{\alpha _{1},\alpha _{1}/2}(\bar{\mathcal {Q}}_{T})\), u satisfies the parabolic condition (1.2), and estimates (2.6) and (2.7) hold.
We next show that for each \(l=1,\ldots,N\), the sequences \(\{A^{l}_{\varepsilon }(x,t)u^{l}_{\varepsilon x}\}\), \(\{B^{l}_{\varepsilon }(x,t,u^{l}_{\varepsilon })u^{l}_{\varepsilon x}\}\), \(\{G_{\varepsilon }^{l}(x,t,\mathbf {u}_{\varepsilon })\}\) converge in \(L^{2}(Q_{T})\) to \(A^{l}(x,t)u^{l}_{x}\), \(B^{l}(x,t,u^{l}) u^{l}_{x}\) and \(G^{l}(x,t,\mathbf {u})\), respectively. Since
$$\begin{aligned} &\operatorname{mes} \bigl\{ (x, t): \min \bigl(\varphi ^{i}_{\varepsilon }(t), \varphi ^{i}(t) \bigr)\leq x\leq\max \bigl(\varphi ^{i}_{\varepsilon }(t)+ \varepsilon , \varphi ^{i}(t)+\varepsilon \bigr), 0\leq t\leq T \bigr\} \\ &\quad \to0 \quad\mbox{as }\varepsilon \to0, i=1,\ldots,n, \end{aligned}$$
and
$$\begin{aligned} \bigl(u^{l}_{\varepsilon x} \bigr)^{2}= \bigl(u^{l}_{\varepsilon x}-u^{l}_{x}+u^{l}_{x} \bigr)^{2}\leq2 \bigl(u^{\varepsilon }_{lx}-u^{l}_{x} \bigr)^{2}+2 \bigl(u^{l}_{x} \bigr)^{2}, \end{aligned}$$
then it follows from (2.2), (3.3) and Lebesgue dominated convergence theorem that
$$\begin{aligned} &\iint_{Q_{T}} \bigl[B^{l}_{\varepsilon }\bigl(x,t,u^{l}_{\varepsilon }\bigr)u^{l}_{\varepsilon x}-B^{l} \bigl(x,t,u^{l} \bigr) u^{l}_{x} \bigr]^{2}\,\mathrm{d}x\,\mathrm{d}t \\ &\quad =\int_{0}^{T} \biggl\{ \int _{0}^{\min{(\varphi ^{l}_{\varepsilon },\varphi ^{l}})} \bigl[\hat {b}\bigl(u^{l}_{\varepsilon }\bigr) u^{l}_{\varepsilon x}-\hat {b}\bigl(u^{l} \bigr)u^{l}_{x} \bigr]^{2} \\ &\qquad{}+\int ^{d}_{\max(\varphi ^{l}_{\varepsilon }+\varepsilon , \varphi ^{l}+\varepsilon )} \bigl[\tilde {b}\bigl(u^{l}_{\varepsilon }\bigr) u^{l}_{\varepsilon x}-\tilde {b}\bigl(u^{l} \bigr)u^{l}_{x} \bigr]^{2} \\ &\qquad{} +\int_{\min(\varphi ^{l}_{\varepsilon },\varphi ^{l})} ^{\max(\varphi ^{l}_{\varepsilon }+\varepsilon , \varphi ^{l}+\varepsilon )} \bigl[B^{l}_{\varepsilon }\bigl(x,t,u^{l}_{\varepsilon }\bigr)u^{l}_{\varepsilon x}-B^{l} \bigl(x,t,u^{l} \bigr) u^{l}_{x} \bigr]^{2}\,\mathrm{d}x \biggr\} \,\mathrm{d}t \\ &\quad \leq C\iint_{Q_{T}} \bigl(u^{l}_{\varepsilon }-u^{l} \bigr)^{2} \bigl(1+\bigl|u^{l}_{x}\bigr| \bigr)^{2} \,\mathrm{d}x\,\mathrm{d}t+C\iint_{Q_{T}} \bigl(u^{l}_{\varepsilon x}-u^{l}_{x} \bigr)^{2}\,\mathrm{d}x\,\mathrm{d}t \\ &\qquad{} +C\int_{0}^{T}\int_{\min(\varphi ^{l}_{\varepsilon },\varphi ^{l})} ^{\max(\varphi ^{l}_{\varepsilon }+\varepsilon , \varphi ^{l}+\varepsilon )} \bigl[ \bigl(u^{l}_{\varepsilon x }-u^{l}_{x} \bigr)^{2}+ \bigl(u^{l}_{x} \bigr)^{2}+1 \bigr]\,\mathrm{d}x\,\mathrm{d}t \\ &\quad \to0 \quad\mbox{as } \varepsilon \to0, l\in\{1,\ldots,n\}, \end{aligned}$$
and
$$\begin{aligned} &\iint_{Q_{T}} \bigl[B^{l}_{\varepsilon }\bigl(x,t,u^{l}_{\varepsilon }\bigr)u^{l}_{\varepsilon x}-B^{l} \bigl(x,t,u^{l} \bigr) u^{l}_{x} \bigr]^{2}\,\mathrm{d}x\,\mathrm{d}t \\ &\quad =\iint_{Q_{T}} \bigl[b^{l} \bigl(u^{l}_{\varepsilon }\bigr)u^{l}_{\varepsilon x}-b^{l} \bigl(u^{l} \bigr) u^{l}_{x} \bigr]^{2}\,\mathrm{d}x\, \mathrm{d}t \\ &\quad \leq C\iint_{Q_{T}} \bigl(u^{l}_{\varepsilon }-u^{l} \bigr)^{2} \bigl(1+\bigl|u^{l}_{x}\bigr| \bigr)^{2} \,\mathrm{d}x\,\mathrm{d}t+C\iint_{Q_{T}} \bigl(u^{l}_{\varepsilon x}-u^{l}_{x} \bigr)^{2}\,\mathrm{d}x\,\mathrm{d}t \\ &\quad \to0 \quad\mbox{as } \varepsilon \to0, l\in\{n+1,\ldots,N\}, \end{aligned}$$
and from (3.10) that
$$\begin{aligned} &\iint_{Q_{T}} \bigl[G_{\varepsilon }^{l}(x,t, \mathbf {u}_{\varepsilon })-G^{l}(x,t,\mathbf {u}) \bigr]^{2}\,\mathrm{d}x\, \mathrm{d}t \\ &\quad\leq \sum_{k=1}^{n+1} \iint_{Q_{T}} \bigl[g^{l}_{k}(\mathbf {u}_{\varepsilon }) -g^{l}_{k}( \mathbf {u}) \bigr]^{2}\,\mathrm{d}x\, \mathrm{d}t \\ &\qquad{} +\sum_{k=1}^{n}\int _{0}^{\tau}\int_{\min\{\varphi ^{k}_{\varepsilon },\varphi ^{k}\}}^{\max\{\varphi ^{k}_{\varepsilon }+\varepsilon ,\varphi ^{k}+\varepsilon \}} \bigl[G_{\varepsilon }^{l}(x,t, \mathbf {u}_{\varepsilon }) -G^{l}(x,t,\mathbf {u}) \bigr]^{2}\,\mathrm{d}x\,\mathrm{d}t \\ &\quad \leq C\iint_{Q_{T}}|\mathbf {u}_{\varepsilon } -\mathbf {u}|^{2}\, \mathrm{d}x\,\mathrm{d} t+C\sum_{k=1}^{n} \int_{0}^{\tau}\int_{\min\{\varphi ^{k}_{\varepsilon },\varphi ^{k}\}}^{\max\{\varphi ^{k}_{\varepsilon }+\varepsilon ,\varphi ^{k}+\varepsilon \}}1 \,\mathrm{d}x\,\mathrm{d}t \\ &\quad \to0,\quad l\in\{1,\ldots,N\}. \end{aligned}$$
The similar argument shows that \(\|A^{l}_{\varepsilon }(x,t)u^{l}_{\varepsilon x}-A^{l}(x,t) u^{l}_{x}\|_{L^{2}(Q_{T})}\to0\) for each \(l=1,\ldots,N\).
Based on the above arguments for sequences \(\{A^{l}_{\varepsilon }(x,t)u^{l}_{\varepsilon x}\}\), \(\{B^{l}_{\varepsilon }(x,t,u^{l}_{\varepsilon })u^{l}_{\varepsilon x}\}\), \(\{G_{\varepsilon }^{l}(x,t,\mathbf {u}_{\varepsilon })\}\) and \(\{u^{l}_{\varepsilon t}\}\), by letting \(\varepsilon \to0\), we conclude from (3.36) that (2.4) holds.
We also see from (3.20) that there exists a subsequence \(\{\mathbf {u}_{\varepsilon '}\}\) (denoted by \(\{\mathbf {u}_{\varepsilon }\}\) still) such that for each \(l=1,\ldots,N\), \(\{(A_{\varepsilon }(x,t)u^{l}_{\varepsilon x})_{x}\}\) converge weakly in \(L^{2}(Q_{T})\) to \(\varpi^{l}(x,t)\). Recalling that \(A_{\varepsilon }(x,t)u^{l}_{\varepsilon x}\to A(x,t) u^{l}_{x}\) in \(L^{2}(Q_{T})\), we deduce \(\varpi_{l}=(A(x,t) u^{l}_{x})_{x}\). These, together with (3.20), imply that \(u^{l}_{xx}\in L^{2}(Q^{l-}_{T})\cap L^{2}(Q^{l+}_{T})\) for each \(l=1,\ldots,n\), \(u^{l}_{xx}\in L^{2}(Q_{T})\) for each \(l=n+1,\ldots,N\), \(A^{l}(x,t) u^{l}_{x}\in V_{2}(Q_{T})\) for each \(l=1,\ldots,N\), and (2.8) holds. Thus (2.4) implies that u satisfies the equations in (1.1a) and (1.1b) for almost all \((x,t)\in Q_{T}\) and the inner boundary condition (1.3) for almost all \(t\in[0,T]\) (see [17, Chapter 3, Section 13]). As we have done in the derivation of (3.21), estimate (2.8) yields (2.9).
For fixed \(l\in\{1,\ldots,N\}\) and \(k\in\{1,\ldots,n+1\}\), \(u^{l}\) satisfies the linear equation
$$ u^{l}_{t}-\underline{a}^{l}u^{l}_{xx}= \underline{b}^{l}(x,t)u^{l}_{x}+\underline {g}^{l}(x,t) \quad \bigl((x,t)\in Q_{k,T}\bigr), $$
where
$$ \underline{a}^{l}:=\left \{ \begin{array}{@{}l} \hat {a}^{l}\quad k\leq l, l=1,\ldots,n,\\ \tilde {a}^{l}\quad k> l, l=1,\ldots,n,\\ a^{l}\quad l=n+1,\ldots,N, \end{array} \right . \qquad \underline{b}^{l}(x,t):=\left \{ \begin{array}{@{}l} \hat {b}^{l}(u^{l}(x,t))\quad k\leq l, l=1,\ldots,n,\\ \tilde {b}^{l}(u^{l}(x,t))\quad k> l, l=1,\ldots,n,\\ b^{l}(u^{l}(x,t))\quad l=n+1,\ldots,N, \end{array} \right . $$
and
$$ \underline{g}^{l}(x,t):=g^{l}_{k} \bigl(\mathbf {u}(x,t) \bigr). $$
Then for any subdomains \(Q'\) and \(Q''\) satisfying \(\bar{Q}\subset Q''\) and \(\bar{Q}''\subset Q_{k,T}\), we have \(\underline{b}^{l}(x,t), \underline{g}^{l}(x,t)\in C^{\alpha _{1},\alpha _{1}/2}(\bar{Q}'')\). The parabolic regularity theory shows that \(u^{l}\in C^{2+\alpha _{1},1+\alpha _{1}/2}(\bar{Q}')\). Hence u satisfies pointwise the equations in (1.1a) and (1.1b) for \((x,t)\in Q_{k,T}\). Consequently, u is a solution in of problem (1.1a)-(1.3) and estimates (2.6)-(2.9) hold.
Step 2. In what follows, we will show that the solution in for problem (1.1a)-(1.3) is unique and estimates (2.10) and (2.11) hold.
Let \(\mathbf {u}_{1}\), \(\mathbf {u}_{2}\) be the solutions in corresponding to \(\boldsymbol {\varphi }_{1}\) and \(\boldsymbol {\varphi }_{2}\), respectively. Set \(\mathbf {w}=\mathbf {u}_{1}-\mathbf {u}_{2}\). Then . We choose \(\boldsymbol {\eta }=\mathbf {w}\) in (2.4) to find
$$\begin{aligned} &\iint_{Q_{\tau}}u^{l}_{it}w^{l}\, \mathrm{d}x\,\mathrm{d}t+ \int_{0}^{\tau}\biggl\{ \int _{0}^{\varphi ^{l}_{i}(t)} \bigl[\hat {a}^{l} u^{l}_{ix}w^{l}_{x}- \hat {b}^{l} \bigl(u^{l}_{i} \bigr)u^{l}_{ix}w^{l} \bigr]\,\mathrm{d}x \\ &\qquad{} +\int^{d}_{\varphi ^{l}_{i}(t)} \bigl[\tilde {a}^{l} u^{l}_{ix}w^{l}_{x}-\tilde {b}^{l} \bigl(u^{l}_{i} \bigr)u^{l}_{ix}w^{l} \bigr]\,\mathrm{d}x \biggr\} -\sum_{k=1}^{n+1} \int_{0}^{\tau}\int_{\varphi ^{k-1}_{i}(t)}^{\varphi ^{k}_{i}(t)}g^{l}_{k}( \mathbf {u}_{i})w^{l}\,\mathrm{d}x\,\mathrm{d}t \\ &\quad =0,\quad i=1,2, l=1,\ldots,n, \end{aligned}$$
and
$$\begin{aligned} & \iint_{Q_{\tau}} \bigl[u^{l}_{it}w^{l}+a^{l} u^{l}_{ix}w^{l}_{x}-b^{l} \bigl(u^{l}_{i} \bigr)u^{l}_{ix}w^{l} \bigr]\,\mathrm{d}x\,\mathrm{d}t-\sum_{k=1}^{n+1} \int_{0}^{\tau}\int_{\varphi ^{k-1}_{i}(t)}^{\varphi ^{k}_{i}(t)}g^{l}_{k}( \mathbf {u}_{i})w^{l}\,\mathrm{d}x\,\mathrm{d}t \\ &\quad =0,\quad i=1,2, l=n+1,\ldots,N. \end{aligned}$$
For each \(l=1,\ldots,N\), by a subtraction of the above equations for \(i=1,2\), we conclude that
$$\begin{aligned} & \frac{1}{2}\int_{0}^{d} \bigl(w^{l}(x,\tau) \bigr)^{2}\,\mathrm{d}x+\int _{0}^{\tau}\biggl\{ \int_{0}^{\varphi ^{l}_{1}} \bigl[\hat {a}^{l} \bigl(w^{l}_{x} \bigr)^{2}- \bigl(\hat {b}^{l} \bigl(u^{l}_{1} \bigr)u^{l}_{1x}- \hat {b}^{l} \bigl(u^{l}_{2} \bigr)u^{l}_{2x} \bigr)w^{l} \bigr]\,\mathrm{d}x \\ &\qquad{} +\int_{\varphi ^{l}_{1}}^{d} \bigl[\tilde {a}^{l} \bigl(w^{l}_{x} \bigr)^{2}- \bigl( \tilde {b}^{l} \bigl(u^{l}_{1} \bigr)u^{l}_{1x}- \tilde {b}^{l} \bigl(u^{l}_{2} \bigr)u^{l}_{2x} \bigr)w^{l} \bigr]\,\mathrm{d}x \\ &\qquad{} +\int_{\varphi ^{l}_{1}}^{\varphi ^{l}_{2}} \bigl(\hat {b}^{l} \bigl(u^{l}_{2} \bigr)-\tilde {b}^{l} \bigl(u^{l}_{2} \bigr) \bigr)u^{l}_{2x}w^{l} \,\mathrm{d}x \biggr\} \,\mathrm{d}t \\ &\quad =\sum_{k=1}^{n+1}\int _{0}^{\tau}\biggl\{ \int_{\varphi ^{k-1}_{2}(t)}^{\varphi ^{k}_{2}(t)} \bigl[g^{l}_{k}(\mathbf {u}_{1})-g^{l}_{k}( \mathbf {u}_{2}) \bigr]w^{l}\,\mathrm{d}x \\ &\qquad{} + \biggl(\int_{\varphi ^{k-1}_{1}(t)}^{\varphi ^{k-1}_{2}(t)}+\int _{\varphi ^{k}_{2}(t)}^{\varphi ^{k}_{1}(t)} \biggr)g^{l}_{k}( \mathbf {u}_{1})w^{l}\,\mathrm{d}x \biggr\} \,\mathrm{d}t,\quad l\in \{1, \ldots,n\}, \end{aligned}$$
and
$$\begin{aligned} & \frac{1}{2}\int_{0}^{d} \bigl(w^{l}(x,\tau) \bigr)^{2}\,\mathrm{d}x+ \iint_{Q_{\tau }} \bigl[a^{l} \bigl(w^{l}_{x} \bigr)^{2}+ \bigl(b^{l} \bigl(u^{l}_{1} \bigr)u^{l}_{1x}-b^{l} \bigl(u^{l}_{2} \bigr)u^{l}_{2x} \bigr)w^{l} \bigr]\,\mathrm{d}x \\ &\quad =\sum_{k=1}^{n+1}\int _{0}^{\tau}\biggl\{ \int_{\varphi ^{k-1}_{2}(t)}^{\varphi ^{k}_{2}(t)} \bigl[g^{l}_{k}(\mathbf {u}_{1})-g^{l}_{k}( \mathbf {u}_{2}) \bigr]w^{l}\,\mathrm{d}x \\ &\qquad{} + \biggl(\int_{\varphi ^{k-1}_{1}(t)}^{\varphi ^{k-1}_{2}(t)}+\int _{\varphi ^{k}_{2}(t)}^{\varphi ^{k}_{1}(t)} \biggr)g^{l}_{k}( \mathbf {u}_{1})w^{l}\,\mathrm{d}x \biggr\} \,\mathrm{d}t,\quad l\in \{n+1, \ldots,N\}. \end{aligned}$$
Then
$$\begin{aligned} & \int_{0}^{d} \bigl(w^{l}(x,\tau) \bigr)^{2}\,\mathrm{d}x+ \iint_{Q_{\tau }} \bigl(w^{l}_{x} \bigr)^{2}\,\mathrm{d}x\,\mathrm{d}t \\ &\quad\leq C\iint_{Q_{\tau}} \bigl[\bigl|w^{l}_{x}w^{l}\bigr|+\bigl|u^{l}_{1x}\bigr| \bigl(w^{l} \bigr)^{2} \bigr]\,\mathrm{d}x\,\mathrm{d}t \\ &\qquad{} +C \biggl|\int_{0}^{\tau}\int _{\varphi ^{l}_{1}(t)}^{\varphi ^{l}_{2}(t)}\bigl|u^{l}_{2x}\bigr|\bigl|w^{l}\bigr| \,\mathrm{d}x\,\mathrm{d}t \biggr|+ \sum_{k=1}^{n+1} \iint_{Q_{\tau}}\bigl|g^{l}_{k}(\mathbf {u}_{1}) -g^{l}_{k}(\mathbf {u}_{2})\bigr|\bigl|w^{l}\bigr|\,\mathrm{d}x \,\mathrm{d}t \\ &\qquad{} +C\sum_{k=1}^{n}\int _{0}^{\tau}\int_{\varphi ^{k}_{1}(t)}^{\varphi ^{k}_{2}(t)}\bigl|w^{l}\bigr| \,\mathrm{d}x\,\mathrm{d}t \\ &\quad \leq\epsilon\iint_{Q_{\tau}}\bigl|w^{l}_{x}\bigr|^{2} \,\mathrm{d}x\,\mathrm{d}t+C(\epsilon)\iint_{Q_{\tau}} \Biggl[1+\sum _{j=1}^{2} \bigl(u^{l}_{jx} \bigr)^{2} \Biggr]|\mathbf {w}|^{2}\,\mathrm{d}x\,\mathrm{d}t \\ &\qquad{} +C\sum_{k=1}^{n}\int _{0}^{\tau}\bigl|\varphi ^{k}_{1}(t)-\varphi ^{k}_{2}(t)\bigr|\,\mathrm{d}t,\quad l=1,\ldots,N. \end{aligned}$$
Setting \(\epsilon=1/2\) and summing the above inequalities with respect to l from \(l=1\) to \(l=N\), we have
$$\begin{aligned} & \int_{0}^{d}\bigl|\mathbf {w}(x,\tau)\bigr|^{2}\, \mathrm{d}x+ \iint_{Q_{\tau}}|\mathbf {w}_{x}|^{2}\,\mathrm{d}x \,\mathrm{d}t \\ &\quad \leq C\iint_{Q_{\tau}} \Biggl[1+\sum_{j=1}^{2}| \mathbf {u}_{jx}|^{2} \Biggr]|\mathbf {w}|^{2}\,\mathrm{d}x\, \mathrm{d}t+C\| \boldsymbol {\varphi }_{1}-\boldsymbol {\varphi }_{2}\|_{L^{2}(0,T)} \\ &\quad \leq C\int_{0}^{\tau}\Biggl[ \Biggl(1+\sum _{j=1}^{2}\|\mathbf {u}_{\varepsilon _{j}x}\| ^{2}_{L^{\infty}(0,d)} \Biggr)\int_{0}^{d} |\mathbf {w}|^{2}\,\mathrm{d}x \Biggr]\,\mathrm{d}t+C\|\boldsymbol {\varphi }_{1}- \boldsymbol {\varphi }_{2} \|_{L^{2}(0,T)}. \end{aligned}$$
Again by Gronwall’s inequality we deduce (2.10), which, together with [17, Chapter 2, formula (3.8)], gives (2.11). Therefore the solution in for problem (1.1a)-(1.3) associated with \(\boldsymbol {\varphi }(t)\) is unique.
Step 3. For \(\boldsymbol {\varphi }\in \mathbf {C}^{2}((0,T])\), we discuss the regularity of u.
For any fixed \(l\in\{1,\ldots,n\}\), let
$$\begin{aligned}& y=x-\varphi ^{l}(t),\qquad t'=t \quad(\mbox{denoted by } t \mbox{ still}), \\& v(y, t)=u^{l} \bigl(y+\varphi ^{l}(t),t \bigr). \end{aligned}$$
Then v satisfies
$$\begin{aligned}& \begin{aligned}[b] v_{t}&=\mathcal{H}(y,t,v) \\ &:=\left \{ \begin{array}{@{}l} \hat {a}^{l}v_{yy}+v_{y}\varphi ^{l}_{t}+\hat {b}^{l}(v)v_{y}+g^{l}_{l}(\mathbf {u}(y+\varphi ^{l}(t),t)) \quad((x,t)\in (-\delta,0)\times(0,T]),\\ \tilde {a}^{l} v_{yy}+v_{y}\varphi ^{l}_{t}+\tilde {b}^{l}(v)v_{y}+g^{l}_{l+1}(\mathbf {u}(y+\varphi ^{l}(t),t)) \quad((x,t)\in (0,\delta)\times(0,T]), \end{array} \right . \end{aligned} \\& v^{-}(0,t)=v^{+}(0,t), \qquad \hat {a}^{l}v^{-}_{y}(0,t)= \tilde {a}^{l}v^{+}_{y}(0,t) \quad \bigl(t\in[0,T] \bigr). \end{aligned}$$
We will use the result of [18] to obtain the regularity of v. To do this, we need the estimate of \(\operatorname{ess\,sup}_{t_{1}\leq t\leq T}\int_{-\delta/2}^{\delta/2}(v_{t})^{2}\, \mathrm{d}y\) for any fixed \(t_{1}\in(0,T)\). Let \(\xi=\xi(y,t)\) be a smooth function with values between 0 and 1 such that \(\xi=0\) for \(|y|\geq \delta\) or \(t\leq t_{1}/2\), \(\xi=1\) for \((y,t)\in[-\delta/2,\delta/2 ]\times[t_{1},T]\) and \(|\xi_{y}|+|\xi_{t}|\leq C(1/\delta,1/t_{1})\), and let
$$ v_{(t)}= \bigl[v(x,t+\Delta t)-v(x,t) \bigr]/\Delta t. $$
For any small enough Δt, consider the equality \(\int_{t_{1}/2}^{\tau} \int_{-\delta}^{\delta}[v_{t}-\mathcal {H}(y,t,v)]_{(t)}v_{(t)}(\xi)^{2}\,\mathrm{d}x\,\mathrm{d}t\). By employing the formula of integration by parts and [17, Chapter II, formula (4.7)], we get
$$\begin{aligned} 0 =& \frac{1}{2}\int_{-\delta}^{\delta} \bigl(v_{(t)}(y,\tau)\xi(y,\tau) \bigr)^{2}\,\mathrm{d}y-\int _{t_{1}/2}^{\tau} \int_{-\delta}^{\delta}(v_{(t)})^{2} \xi\xi_{t}\,\mathrm{d}y\,\mathrm{d}t \\ &{} +\int_{t_{1}/2}^{\tau} \biggl\{ \int _{-\delta}^{0} \bigl[ \bigl(\hat {a}^{l}(v_{y(t)} \xi)^{2}+2\hat {a}^{l}v_{y(t)}v_{(t)}\xi \xi_{y} \bigr)+ \bigl( \bigl(\varphi ^{l}_{t} \bigr)_{(t)}v_{y}+\varphi _{t}(t+\Delta t)v_{y(t)} \bigr)v_{(t)}(\xi)^{2} \\ &{} + \bigl( \bigl(\hat {b}^{l}(v) \bigr)_{(t)}v_{y}+ \hat {b}^{l} \bigl(v(y,t+\Delta t) \bigr)v_{y(t)} \bigr)v_{(t)}(\xi)^{2}+ \bigl(g^{l}_{l} \bigl(\mathbf {u}\bigl(y+\varphi ^{l}(t),t \bigr) \bigr) \bigr)_{(t)}v_{(t)}( \xi)^{2} \bigr]\,\mathrm{d}y \\ &{} +\int_{0}^{\delta} \bigl[ \bigl( \tilde {a}^{l}(v_{y(t)}\xi)^{2}+2\tilde {a}^{l}v_{y(t)}v_{(t)} \xi\xi_{y} \bigr)+ \bigl( \bigl(\varphi ^{l}_{t} \bigr)_{(t)}v_{y}+\varphi _{t}(t+\Delta t)v_{y(t)} \bigr)v_{(t)}(\xi)^{2} \\ &{} + \bigl( \bigl(\tilde {b}^{l}(v) \bigr)_{(t)}v_{y}+ \tilde {b}^{l} \bigl(v(y,t+\Delta t) \bigr)v_{y(t)} \bigr)v_{(t)}(\xi)^{2} \\ &{}+ \bigl(g^{l}_{l+1} \bigl(\mathbf {u}\bigl(y+ \varphi ^{l}(t),t \bigr) \bigr) \bigr)_{(t)}v_{(t)}( \xi)^{2} \bigr]\,\mathrm{d}y \biggr\} \,\mathrm{d}t, \end{aligned}$$
where \(v_{y(t)}=(v_{y})_{(t)}\). Some tedious computation and Cauchy’s inequality yield
$$\begin{aligned} &\frac{1}{2}\int_{-\delta}^{\delta} \bigl(v_{(t)}(y,\tau)\xi(y,\tau) \bigr)^{2}\,\mathrm{d}y+ \nu_{0}\int_{t_{1}/2}^{\tau} \int _{-\delta}^{\delta}(v_{y(t)}\xi)^{2}\, \mathrm{d} y\,\mathrm{d}t \\ &\quad\leq\epsilon\int_{t_{1}/2}^{\tau} \int _{-\delta}^{\delta }(v_{y(t)}\xi)^{2} \, \mathrm{d}y \,\mathrm{d}t \\ &\qquad{} +C \bigl(\epsilon,1/\delta,1/t_{1},\bigl\| \varphi ^{l}\bigr\| _{C^{2}([t_{1}/2,T])} \bigr)\int_{t_{1}/2}^{\tau} \int _{-\delta}^{\delta} \bigl[1+(v_{y})^{2}+(v_{(t)})^{2}+(v_{(t)} \xi)^{2}v_{y} \bigr]\,\mathrm{d}y\,\mathrm{d}t \\ &\qquad{} + C \bigl(\bigl\| \varphi ^{l}\bigr\| _{C^{1}([t_{1}/2,T])} \bigr)\sum _{i=1}^{N}\iint_{Q_{T}} \bigl[ \bigl(u^{i}_{x}(x,t) \bigr)^{2}+ \bigl(u^{i}_{t}(x,t) \bigr)^{2} \bigr]\,\mathrm{d}x \,\mathrm{d}t. \end{aligned}$$
We choose \(\epsilon=\nu_{0}/4\) and employ (2.8) to find
$$\begin{aligned} & \int_{-\delta}^{\delta} \bigl(v_{(t)}(y,\tau) \xi(y,\tau) \bigr)^{2}\,\mathrm{d}y+\int_{t_{1}/2}^{\tau} \int_{-\delta}^{\delta}(v_{y(t)}\xi )^{2}\,\mathrm{d}y\,\mathrm{d}t \\ &\quad \leq C \bigl(1/\delta,1/t_{1},\bigl\| \varphi ^{l} \bigr\| _{C^{2}([0,T])} \bigr) \biggl\{ 1+\int_{t_{1}/2}^{\tau } \int_{-\delta}^{\delta} (v_{(t)} \xi)^{2}v_{y} \,\mathrm{d}y\,\mathrm{d}t \biggr\} \\ &\quad \leq C \bigl(1/\delta,1/t_{1},\bigl\| \varphi ^{l} \bigr\| _{C^{2}([t_{1}/2,T])} \bigr) \biggl\{ 1+\int_{t_{1}/2}^{\tau} \|v_{y} \|_{L^{\infty}(-\delta,\delta)}\int_{-\delta}^{\delta} (v_{(t)} \xi)^{2}\,\mathrm{d}y\,\mathrm{d}t \biggr\} . \end{aligned}$$
As we have done in the derivation of (3.20), by Gronwall’s inequality we get
$$\begin{aligned} \int_{-\delta}^{\delta} \bigl(v_{(t)}(y,\tau) \xi(y,\tau) \bigr)^{2}\,\mathrm{d}y+\int_{t_{1}/2}^{\tau} \int_{-\delta}^{\delta}(v_{y(t)}\xi )^{2}\,\mathrm{d}y\,\mathrm{d}t\leq C \bigl(1/\delta,1/t_{1}, \bigl\| \varphi ^{l}\bigr\| _{C^{2}([0,T])} \bigr). \end{aligned}$$
Consequently,
$$\begin{aligned} \int_{-\delta/2}^{\delta/2}(v_{(t)})^{2} \,\mathrm{d}y+\int_{t_{1}}^{\tau} \int _{-\delta/2}^{\delta/2}(v_{y(t)})^{2} \, \mathrm{d}y\,\mathrm{d}t\leq C \bigl(1/\delta,1/t_{1},\bigl\| \varphi ^{l} \bigr\| _{C^{2}([t_{1}/2,T])} \bigr). \end{aligned}$$
By [17, Chapter II, Lemma 4.11], this inequality implies that
$$\begin{aligned} \mathop{\operatorname{ess\,sup}}\limits_{t_{1}\leq t\leq T}\int_{-\delta/2}^{\delta/2}(v_{t})^{2} \,\mathrm{d}y+\int_{t_{1}}^{T} \int _{-\delta/2}^{\delta/2}(v_{yt})^{2} \, \mathrm{d}y\,\mathrm{d}t\leq C \bigl(1/\delta,1/t_{1},\bigl\| \varphi ^{l} \bigr\| _{C^{2}([t_{1}/2,T])} \bigr). \end{aligned}$$
(3.37)
Hence \(u^{l}_{xt}\in\L^{2}(Q^{l}(\delta,t_{1}))\). Using (3.37), hypothesis (H)(iii) and [18, Theorem 1.1], we deduce that \(v_{y}(y,t)\) is continuous with respect to y in \([-\delta/4,0]\) and in \([0, \delta/4]\) for almost all \(t\in[t_{1},T]\), and \(v_{t}\) is continuous in \([-\delta/4,\delta/4]\times[t_{1},T]\). Since \(u^{l}_{x}(x,t)=v_{y}(x-\varphi ^{l}(t),t)\) and \(u^{l}_{t}(x,t)=v_{t}(x-\varphi ^{l}(t),t)-v_{y}(x-\varphi ^{l}(t),t)\varphi ^{l}_{t}(t)\), then for almost all \(t\in[t_{1},T]\), \(u^{l}_{x}(x,t)\) and \(u^{l}_{t}(x,t)\) are continuous with respect to x in \([\varphi ^{l}(t)-\delta/4,\varphi ^{l}(t)]\) and in \([\varphi ^{l}(t),\varphi ^{l}(t)+\delta/4]\). □
The following corollary follows directly from Theorem 2.1.
Corollary 3.3
Assume that
\(\boldsymbol {\varphi }_{m}(t)\in \mathbf {C}^{2}([0,T])\), \(m=1,2,\ldots\) , satisfy (2.5) and the sequence
\(\{\boldsymbol {\varphi }_{m}(t)\}\)
converges in
\(\mathbf {W}^{1}_{2}(0,T)\)
to
\(\boldsymbol {\varphi }(t)\). If
\(\mathbf {u}_{m}\), u
are the solutions in
of (1.1a)-(1.3) corresponding to
\(\boldsymbol {\varphi }_{m}\)
and
φ, respectively, then there exists a subsequence (we retain the same notation for it) \(\{\mathbf {u}_{m}\}\)
such that
$$\begin{aligned} \mathbf {u}_{m}\to \mathbf {u}\quad\textit{in }\mathcal{C}(\bar{Q}_{T}), \qquad \mathbf {u}_{m t}\rightharpoonup \mathbf {u}_{t} \quad\textit{weakly in } \mathcal {L}^{2}(Q_{T}),\qquad \mathbf {u}_{m}\to \mathbf {u}\quad \textit{in } \mathcal{V}_{2}^{1, 0}(\mathcal{Q}_{T}). \end{aligned}$$