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Existence and Comparison Results for Non-Uniformly Parabolic Problems

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Abstract

In this paper we prove existence and comparison results for nonlinear parabolic equations which are modeled on the problem

$$\left\{\begin{array}{ll}{u_t - {\rm div}\,\left(\frac{1}{(1+|u|)^{\alpha}}|Du|^{p-2}Du\right) =f\quad\hskip 2pt \,\,{\rm in}\,\Omega\times(0,T),}\\ {u=0\qquad\qquad\qquad\qquad\quad\quad\qquad{\rm on}\,\partial\Omega\times(0,T),}\\ {u(x,0)=u_0(x)\quad\qquad\qquad\qquad\qquad{\rm in}\,\Omega,}\end{array}\right.$$

where T > 0, Ω is a bounded open set in \({\mathbb{R}^n, n \ge 2, 1 < p < n, \alpha \ge 0, f \in L^{\infty}(0, T;L^q(\Omega))}\), with q > n/p, and u 0 is a bounded function.

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Authors and Affiliations

  1. Dipartimento S.A.V.A., Facoltà di Ingegneria, Via Duca degli Abruzzi, 86100, Termoli (CB), Italy

    Francesco Della Pietra

  2. Dipartimento di Matematica, Seconda Università degli Studi di Napoli, Via Vivaldi 43, 81100, Caserta, Italy

    Giuseppina di Blasio

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  1. Francesco Della Pietra
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  2. Giuseppina di Blasio
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Correspondence to Francesco Della Pietra.

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Della Pietra, F., di Blasio, G. Existence and Comparison Results for Non-Uniformly Parabolic Problems. Mediterr. J. Math. 7, 323–340 (2010). https://doi.org/10.1007/s00009-010-0066-8

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  • DOI: https://doi.org/10.1007/s00009-010-0066-8

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