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arXiv · 2204.09307

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

Razvan Gabriel Iagar·Philippe Laurençot·Ariel Sánchez
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Abstract

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 $(t, x) \in (0,\infty) \times \mathbb{R}^N$, 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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Razvan Gabriel Iagar, Philippe Laurençot, Ariel Sánchez. 2022-04-20. Self-similar shrinking of supports and non-extinction for a nonlinear diffusion equation with spatially inhomogeneous strong absorption. https://arxiv.org/abs/2204.09307

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