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Nonlinear parabolic problems with a very general quadratic gradient term. (English) Zbl 1212.35204

Summary: We study existence and regularity of distributional solutions for a class of nonlinear parabolic problems. The equations we consider have a quasi-linear diffusion operator and a lower-order term, which may grow quadratically in the gradient and may have a very fast growth (for instance, exponential) with respect to the solution. The model problem we refer to is the following: \(u_t-\Delta u=\beta (u)| \nabla u| ^2+f(x,t)\) in \(\Omega \times (0,T)\), \(u(x,t)=0\) on \(\partial \Omega \times (0,T)\), \(u(x,0)=u_0(x)\) in \(\Omega \), with \(\Omega \subset \mathbb R^N\) a bounded open set, \(T>0\), and \(\beta (u)\sim e^{| u| }\); as far as the data are concerned, we assume \(\exp (\exp (| u_0| ))\in L^2(\Omega )\), and \(f\in X(0,T;Y(\Omega ))\), where \(X\), \(Y\) are Orlicz spaces of logarithmic and exponential type, respectively. We also study a semilinear problem having a superlinear reaction term, a problem that is linked with the above problem by a change of unknown. Likewise, we deal with some other related problems, which include a gradient term and a reaction term together.

MSC:

35K55 Nonlinear parabolic equations
35B65 Smoothness and regularity of solutions to PDEs
35K57 Reaction-diffusion equations
35K20 Initial-boundary value problems for second-order parabolic equations
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