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Optimal bounds and blow up phenomena for parabolic problems in narrowing domains. (English) Zbl 0919.35069
Summary: We consider degenerate parabolic problems in domains with noncompact boundary and infinite volume, in any spatial dimension. The equation is of doubly nonlinear type. On the boundary we prescribe a homogeneous Neumann condition. The spatial domain is narrowing at infinity. We prove uniform convergence to zero of solutions as time approaches infinity. To this end, due to the geometry of the domain, the requirement that the initial datum have finite mass is not enough, and we have to stipulate the further assumption that a certain moment of the initial datum (connected with the geometry of the domain) is finite. We prove optimal asymptotic estimates of the solution. Moreover, we apply our method to the investigation of blow-up problems in narrowing domains, obtaining a sharp condition, in integral form, for the existence of solutions defined for all positive times.

MSC:
35K65 Degenerate parabolic equations
35B45 A priori estimates in context of PDEs
35K60 Nonlinear initial, boundary and initial-boundary value problems for linear parabolic equations
35B40 Asymptotic behavior of solutions to PDEs
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