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Study of a doubly nonlinear heat equation with no growth assumptions on the parabolic term. (English) Zbl 0685.35052
The following problem is considered: For a bounded domain $$\Omega \subset R^ N$$ with Lipschitz boundary, to find a solution $$u\in L^{\infty}(0,T;W_ 0^{1,q}(\Omega))of$$ $(\partial /\partial t)b(u)-div D\phi (\nabla u)=f\quad in\quad \Omega \times (0,T),\quad u=0\quad on\quad \partial \Omega \times (o,T),\quad b(u)|_{t=0}=b(u_ 0).$ The main results of the work are two theorems, Th. 1 and Th. 2, which are proven under the following general hypotheses concerning the structure functions b,$$\phi$$ : Let $$1<q<+\infty$$, $$q>2N/N+2$$, $$r>1$$, $$\alpha >0$$, $$T>0$$. Let b be a locally Lipschitz, monotone (non necessarily strictly) increasing function, $$b(0)=0$$. Let $$\phi$$ be a real valued, convex, $$C^ 1$$ functional on $$[L^ q(\Omega)]^ N$$, such that $$D\phi$$ is bounded on the bounded sets of $$[L^ q(\Omega)]^ N$$, $$\phi (0)=0$$, and $$\phi (w)>\alpha (\int_{\Omega}| w|^ q)^{r/q}$$ for any $$w\in [L^ q(\Omega)]^ N.$$
For Th. 1 the data and forcing term of the problem satisfy $$u_ 0\in W_ 0^{1,q}(\Omega),b(u_ 0)\in L^ 2(\Omega),\quad f\in W^{1,1}(0,T;L^ 2(\Omega)).$$
For Th. 2, instead, $$u_ 0\in W_ 0^{1,q}(\Omega)$$, $$b(u_ 0)\in L^ 1_{loc}(\Omega)\cap W^{-1,q'}(\Omega)$$, $$f\in W^{1,1}(0,T;W^{-1,q'}(\Omega)).$$
The authors stress the fact that no growth condition on b is imposed and that the evolution equation may become stationary in a subdomain of $$\Omega$$ $$\times (0,T).$$
An interesting feature of Th. 1 and Th. 2 is the following comparison statement in both theorems: If $$u_{01}$$ and $$u_{02}$$, $$f_ 1$$ and $$f_ 2$$ satisfy the conditions for Th. 1 (resp. Th. 2), and $$b(u_{01})-b(u_{02})$$ is a.e. positive on $$\Omega$$, $$f_ 1-f_ 2$$ is “positive”, then there exist a solution $$u_ 1$$ associated to $$u_{01},f_ 1$$, and a solution $$u_ 2$$, associated to $$u_{02},f_ 2$$ such that $$b(u_ 1)-b(u_ 2)$$ is a.e. positive on $$\Omega$$ $$\times (0,T)$$. Here “positive” means a.e. positive on $$\Omega$$ $$\times (0,T)$$ in Th. 1, and means $$<(f_ 1-f_ 2)(t),\phi >>0\quad a.e.$$ on (0,T) for any $$\phi \in W_ 0^{1,q}(\Omega)$$ in Th. 2.
For the proofs a Galerkin approximation is used via truncation of b and perturbation to a new (strictly) monotone function $$b^{\eta}(u)+\epsilon u.$$
The references contain 13 items.
Reviewer: J.E.Bouillet

##### MSC:
 35K55 Nonlinear parabolic equations 35K20 Initial-boundary value problems for second-order parabolic equations 35B05 Oscillation, zeros of solutions, mean value theorems, etc. in context of PDEs 35B40 Asymptotic behavior of solutions to PDEs
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