Uniqueness for an elliptic-parabolic problem with Neumann boundary condition. (English) Zbl 1053.35109

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 H. W. Alt and S. Luckhaus [Math. Z. 183, 311–341 (1983; Zbl 0497.35049)], by J. I. Diaz and F. de Thélin [SIAM J. Math. Anal. 25, 1085–1111 (1994; Zbl 0808.35066) and by P. Bénilan and 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 \(L^1(\Omega)\).


35M10 PDEs of mixed type
35K65 Degenerate parabolic equations
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35J65 Nonlinear boundary value problems for linear elliptic equations
47H06 Nonlinear accretive operators, dissipative operators, etc.
47H20 Semigroups of nonlinear operators
35D05 Existence of generalized solutions of PDE (MSC2000)