Uniqueness for an elliptic-parabolic problem with Neumann boundary condition
The authors consider for a smooth domain \(\Omega\subset\mathbb R^N\) the problem \[ b(u)-\Delta u + \text{div}\left(F(u)\right)=f\quad\text{in }\Omega;\qquad(\nabla u -F(u))\cdot\nu=0\quad\text{on }\partial\Omega\tag{1} \] and the corresponding evolution problem \[ \begin{aligned} b(u)_t-\Delta u + \text{div}\left(F(u)\right)&= f\quad\text{in }Q:=\Omega\times (0,T);\\ (\nabla u -F(u))\cdot\nu&= 0\quad\text{on } \partial\Omega\times(0,T);\tag{2}\\ b(u(x,0))&= b^0\quad\text{in }\Omega. \end{aligned} \] Here \(\nu\) denotes the exterior unit normal vector to the boundary of \(\partial\Omega.\) Similar problems have been studied by \textit{H. W. Alt} and \textit{S. Luckhaus} [Math. Z. 183, 311--341 (1983; Zbl 0497.35049)], by \textit{J. I. Diaz} and \textit{F. de Thélin} [SIAM J. Math. Anal. 25, 1085--1111 (1994; Zbl 0808.35066) and by \textit{P. Bénilan} and \textit{P. Wittbold} [Adv. Differ. Equ. 1, 1053--1073 (1996; Zbl 0858.35064)]. Assuming that \(b:\mathbb R \rightarrow \mathbb R\) is increasing with \(b(0)=0\), \(F:\mathbb R\rightarrow \mathbb R^N\) is continuous with \(F(0)=0\) and that the following (growth) conditions (H1) there exist \(c>0\), \(\delta>0\) such that \(| F(z)| ^2\leq c(1+| z| ^2+(zb(z))^{1-\delta})\) for all \(z\in \mathbb R\), (H2) there exists a function \(\beta:\mathbb R^+\rightarrow \mathbb R^+\) such that \(\beta(r)\rightarrow\infty\) as \(r\rightarrow\infty\) and \(| b(z)| \geq\beta(| z| )\) for all \(z\in\mathbb R\), (H3) for all compact sets \(K\subset\mathbb R\) there exists \(\alpha>0\) such that \(F\) is Hölder continuous of order \(\alpha\) on \(K\), hold, the authors establish the existence and uniqueness of a weak solution of (1), where a function \(u\in H^1(\Omega)\) such that \(b(u)\in L^1(\Omega)\) and \(F(u)\in L^2(\Omega)\) is called a weak solution of (1), if \[ \int_\Omega b(u)\xi+\int_\Omega (\nabla u - F(u))\nabla \xi=\int_\Omega f\xi \] for all \(\xi\in H^1(\Omega)\cap L^\infty(\Omega).\) Here \(f\) is assumed to belong to the class \(L^1(\Omega)\). Under the additional assumption that (H4) there exist \(c>0\), \(\delta>0\) such that \(| F(z)| ^2\leq c(1+| z| ^{2(1-\delta)}+zb(z))\) for all \(z\in \mathbb R\), the authors prove the existence and uniqueness of a weak solution of (2). A weak solution of (2) is defined as a function \(u\in L^2(0,T;H^1(\Omega))\) such that \(b(u)\in L^1(Q)\), \(F(u)\in L^2(Q)\) and \[ \int\limits_0^T\int_\Omega (b^0-b(u))\xi_t+\int_0^T\int_\Omega (\nabla u -F(u))\nabla \xi=\int_0^T\int\limits_\Omega f\xi \] is fulfilled for all \(\xi\in L^2(0,T;H^1(\Omega))\cap L^\infty(Q)\) such that \(\xi_t\in L^\infty(Q)\) and \(\xi(T)=0\). For this definition one assumes that \(f\in L^1(Q)\) and \(b^0\) belongs to \(L^1(\Omega)\).
scientific article; zbMATH DE number 2103747