Self-similar asymptotics of solutions to heat equation with inverse square potential

Dominika Pilarczyk

doi:10.1007/s00028-012-0169-8

Abstract

We show that the large-time behavior of solutions to the Cauchy problem for the linear heat equation with the inverse square potential is described by explicit self-similar solutions.

Highlights

In this paper, we study properties of weak solutions to the linear and singular initial value problem ut = λ u + |x|2 u on Rn × (0, ∞) (1.1) u(x, 0) = u0(x), (1.2)where n 3, the parameter λ ∈ R is given and assumptions on the initial condition u0 are stated below.The initial value problem (1.1)–(1.2) was popularized by Baras and Goldstein [1]who discovered the “instantaneous blow up” of solutions, namely, the fact that Cauchy problem (1.1)–(1.2)has no positive local in time solutions if λ >
(1.1)–(1.2) which is closely related to the one by Vázquez and Zuazua [13] and which is based on the estimates of the kernel of the Schrödinger operator H u = −
Let us mention that there is extensive literature on properties of the Schrödinger semigroup of linear operators e−t HV generated by HV ≡ − + V, where a potential V = V (x) is less singular at the origin, for example, when it belongs to the socalled Kato class

Summary

Introduction

We study properties of weak solutions to the linear and singular initial value problem ut =. Where n 3, the parameter λ ∈ R is given and assumptions on the initial condition u0 are stated below. The initial value problem (1.1)–(1.2) was popularized by Baras and Goldstein [1]. Who discovered the “instantaneous blow up” of solutions, namely, the fact that Cauchy problem (1.1)–(1.2). The authors of [1] found necessary and sufficient conditions for u0 so that a nonnegative solution exists. 3, we explain the role of the Hardy inequality (1.3) in the proof of existence of solutions to (1.1)–(1.2). 0, Mathematics Subject Classification (2000): 35K05, 35K15, 35B05, 35B40 Keywords: Heat equation, singular potential, self-similar solutions, large-time asymptotics

Pilarczyk

Main results

Existence of solutions

Properties of fundamental solution