Article Dans Une Revue Communications in Contemporary Mathematics Année : 2023

Self-similar shrinking of supports and non-extinction for a nonlinear diffusion equation with spatially inhomogeneous strong absorption

Résumé

We study the dynamics of the following porous medium equation with strong absorption $$ \partial_t u=\Delta u^m-|x|^{\sigma}u^q, $$ posed for $(t, x) \in (0,\infty) × \mathnn{R}^N$, with $m > 1$, $q \in (0, 1)$ and $\sigma > 2(1 − q)/(m − 1)$. Considering the Cauchy problem with non-negative initial condition $u_0 \in L^\infty(\mathbb{R}^N)$ instantaneous shrinking and localization of supports for the solution u(t) at any t > 0 are established. With the help of this property, existence and uniqueness of a nonnegative compactly supported and radially symmetric forward self-similar solution with algebraic decay in time are proven. Finally, it is shown that finite time extinction does not occur for a wide class of initial conditions and this unique self-similar solution is the pattern for large time behavior of these general solutions.

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hal-03643979 , version 1 (17-04-2022)

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Razvan Gabriel Iagar, Philippe Laurençot, Ariel Sánchez. Self-similar shrinking of supports and non-extinction for a nonlinear diffusion equation with spatially inhomogeneous strong absorption. Communications in Contemporary Mathematics, 2023, 26 (06), pp.2350028. ⟨10.1142/S0219199723500281⟩. ⟨hal-03643979⟩
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