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The paper studies the equation $$ {u_{t}}= \operatorname{div} \bigl(a(x)\vert {\nabla u} \vert ^{p(x) - 2}\nabla u \bigr), $$ with the boundary degeneracy coming from \(a(x)\vert_{x\in \partial \Omega }=0\). The paper introduces a new kind of weak solutions of the equation. One can study the stability of the new kind of weak solutions without any boundary value condition.Keywordselectrorheological fluid equationboundary degeneracywell-posednessMSC35K5535K9235K8535R351 IntroductionConsider the evolutionary \(p(x)\)-Laplacian equation $$ u_{t} =\operatorname{div} \bigl(a(x)\vert \nabla u \vert ^{p(x)-2}\nabla u \bigr),\quad (x,t) \in {Q_{T}} = \Omega \times (0,T), $$ (1.1) which comes from a new interesting kind of fluids: the so-called electrorheological fluids (see [1, 2]). Here, \(\Omega \subset \mathbb{R}^{N}\) is a bounded domain with smooth boundary ∂Ω, \(p(x)\) is a measurable function, we assume that \(a(x)>0, x \in \Omega \), \(a(x)=0, x\in \partial \Omega \). If \(a(x)=1\), the initial boundary value problem of equation (1.1) has been widely studied [3–5]. If \(a(x)|_{ x\in \partial \Omega }=0\), the situation may completely different from that of \(a(x)\equiv 1\). To see that, let us suppose that u and v would be two classical solutions of equation (1.1) with the initial values \(u(x,0)\) and \(v(x,0)\), respectively. It is easy to show that $$ \int_{\Omega } \bigl\vert u(x,t)-v(x,t) \bigr\vert ^{2}\,dx\leq \int_{\Omega } \bigl\vert u_{0}(x)-v_{0}(x) \bigr\vert ^{2}\,dx. $$ (1.2) It implies that the classical solutions (if there are any) of equation (1.1) are controlled by the initial value completely. Certainly, since equation (1.1) is degenerate on the boundary and may be degenerate or singular at points where \(\vert \nabla u \vert =0\), it only has a weak solution generally, so whether the conclusion (1.1) is true or not remains to be verified. This is the main aim of the paper.If \(a(x)=d^{\alpha }(x)\), \(d=\operatorname{dist}(x,\partial \Omega )\) is the distance from the boundary, the well-posedness of the solutions of the equation $$ u_{t} =\operatorname{div} \bigl(d^{\alpha }(x) \vert \nabla u \vert ^{p-2}\nabla u \bigr), \quad (x,t) \in {Q_{T}} , $$ (1.3) was first studied by Yin and Wang [6], and later by Yin and Wang [7], Zhan and Xie [8], etc. While the corresponding equation related to the \(p(x)\)-Laplacian $$ u_{t} =\operatorname{div} \bigl(d^{\alpha }(x) \vert \nabla u \vert ^{p(x)-2}\nabla u \bigr), \quad (x,t) \in{Q_{T}}, $$ (1.4) was studied by Zhan and Wen [9], and Zhan [10].In this short paper, we will study the well-posedness of the solutions of equation (1.1) with the initial value $$ u|_{t=0} = u_{0}(x),\quad x\in \Omega , $$ (1.5) but without any boundary value condition.2 Basic functional space and a new kind of weak solutionLet us recall some definitions and basic properties of the weighted variable exponent Lebesgue spaces \({L^{p(x)}}(a,\Omega )\) and the weighted variable exponent Sobolev spaces \({W^{1,p(x)}}(a,\Omega )\) according to [11]. Set $$ {C_{+} }(\overline{\Omega }) = \Bigl\{ h \in C(\overline{\Omega }): \mathop{\min }_{x \in \overline{\Omega }} h(x)> 1 \Bigr\} . $$ For any \(h \in {C_{+}}(\overline{\Omega} )\) we define $$ {h_{+} } = \mathop{\sup}_{x \in \Omega } h(x), \qquad {h_{-} } = \mathop{\inf }_{x \in \Omega } h(x). $$For any \(p \in {C_{+}}(\overline{\Omega })\), \({L^{p(x)}}(a,\Omega )\) consists of all measurable real-valued functions u such that $$ \int_{\Omega} {a(x) \bigl\vert u(x){ \bigr\vert ^{p(x)}}}\,dx < \infty , $$ endowed with the Luxemburg norm $$ \Vert u \Vert _{{L^{p(x)}}(a,\Omega )} = \inf \biggl\{ \lambda > 0: \int_{\Omega} {a(x) \biggl\vert \frac{{u(x)}}{\lambda }{ \biggr\vert ^{p(x)}}}\,dx \leqslant 1 \biggr\} . $$\({W^{1,p(x)}}(a,\Omega )\) is defined by $$ {W^{1,p(x)}}(a,\Omega ) = \bigl\{ u \in {L^{p(x)}}(\Omega ):\vert \nabla u \vert \in {L^{p(x)}}(a,\Omega ) \bigr\} , $$ endowed with the norm $$ {\Vert u \Vert _{{W^{1,p(x)}}(a,\Omega )}} = {\Vert u \Vert _{{L^{p(x)}}(\Omega )}} + { \Vert {\nabla u} \Vert _{{L^{p(x)}}(a, \Omega )}}. $$ It is easy to see that the norm $$ \bigl\vert {\Vert u \Vert } \bigr\vert = \inf \biggl\{ \mu > 0: \int_{\Omega} \biggl( \biggl\vert \frac{u(x)}{\mu}{ \biggr\vert ^{p(x)}} + a(x) \biggl\vert \frac{\nabla u(x)}{\mu }{ \biggr\vert ^{p(x)}}\,dx \biggr)\leqslant 1 \biggr\} $$ is equivalent to \(\Vert u \Vert _{W^{1,p(x)}(a,\Omega )}\).Lemma 2.1Denote$$ \rho (u) = \int_{\Omega} {a(x)\vert u{ \vert ^{p(x)}}}\,dx \quad \textit{for all } u \in L^{p(x)}(a,\Omega ). $$Then(1)\(\rho (u) > 1\)\(( = 1; < 1)\)if and only if\({\Vert u \Vert _{{L^{p(x)}}(a,\Omega )}} > 1\)\(( = 1; < 1)\), respectively;
