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Parabolic problems with nonlinear boundary conditions and critical nonlinearities. (English) Zbl 0938.35077
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.

##### MSC:
 35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
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##### References:
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