Parabolic problems with nonlinear boundary conditions and critical nonlinearities
The authors study the nonlinear parabolic problem \[ \begin{gathered} \frac {\partial u}{\partial t} =\Delta u + f(u) \text{ in } \Omega\times(0,T), \\ \frac {\partial u}{\partial \nu} = g(u) \text{ on } \partial\Omega\times(0,T), \\ u(\cdot,0) = u_0 \text{ in } \Omega, \end{gathered} \] where \(\Omega \subset \mathbb R^N\) has unit outer normal \(\nu\), and \(u\) depends on \(x\) and \(t\). Their main theorem is that if there are constants \(q \in (1,\infty)\) and \(C\) such that \[ \begin{aligned} |f(u)-f(v)|&\leq C|u-v|(|u|^{2q/N}+|v|^{2q/N} +1),\\ |g(u)-g(v)|&\leq C|u-v|(|u|^{q/N}+|v|^{q/N} +1) \end{aligned} \tag{*} \] (if \(N \geq 2\)), and if \(u_0 \in L^q(\Omega)\), then this problem has a unique solution. (If \(N=1\), the exponents in the inequality for \(g\) must be less than \(q\).) They also show that if \(u_0 \in W^{1,q}\) with \(q \in(1,\infty)\), then the condition (*) can be relaxed. These results include all known results on existence for problems with critical growth. The method of proof relies on various interpolation theorems for scales of Banach spaces.
interpolation theorems for scales of Banach spaces
scientific article; zbMATH DE number 1339375
scientific article