Long-time behavior for the porous medium equation with small initial\n energy

2020/11/09 by Lorenzo Brasco, Brasco, Lorenzo, Bruno Volzone +1
Computer Science · Mathematics · #35B40 #35J61 #35K55 #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #Differential Equations and Numerical Methods #FOS: Mathematics #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2011.04619

openalex publication_date 2020/11/09 · openalex created_date 2022/10/04 · openalex updated_date 2026/07/28

Abstract

We study the long-time behavior for the solution of the Porous Medium\nEquation in an open bounded connected set, with smooth boundary. Homogeneous\nDirichlet boundary conditions are considered. We prove that if the initial\ndatum has sufficiently small energy, then the solution converges to a\nnontrivial constant sign solution of a sublinear Lane-Emden equation, once\nsuitably rescaled. We point out that the initial datum is allowed to be\nsign-changing. We also give a sufficient energetic criterion on the initial\ndatum, which permits to decide whether convergence takes place towards the\npositive solution or to the negative one.\n

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