Study of a doubly nonlinear heat equation with no growth assumptions on the parabolic term. (English) Zbl 0685.35052

Blanchard, Dominique; Francfort, Gilles A.

SIAM J. Math. Anal. 19, No. 5, 1032-1056 (1988).

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

Keywords:

energy estimates; Galerkin method; existence theorem; no growth; condition; comparison statement