# Selfsimilar profiles in large time asymptotics of solutions to damped wave equations

Studia Mathematica (2000)

• Volume: 143, Issue: 2, page 175-197
• ISSN: 0039-3223

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## Abstract

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Large time behavior of solutions to the generalized damped wave equation ${u}_{tt}+A{u}_{t}+\nu Bu+F\left(x,t,u,{u}_{t},\nabla u\right)=0$ for $\left(x,t\right)\in {ℝ}^{n}×\left[0,\infty \right)$ is studied. First, we consider the linear nonhomogeneous equation, i.e. with F = F(x,t) independent of u. We impose conditions on the operators A and B, on F, as well as on the initial data which lead to the selfsimilar large time asymptotics of solutions. Next, this abstract result is applied to the equation where $A{u}_{t}={u}_{t}$, $Bu=-\Delta u$, and the nonlinear term is either $|{u}_{t}{|}^{q-1}{u}_{t}$ or ${|u|}^{\alpha -1}u$. In this case, the asymptotic profile of solutions is given by a multiple of the Gauss-Weierstrass kernel. Our method of proof does not require the smallness assumption on the initial conditions.

## How to cite

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Karch, Grzegorz. "Selfsimilar profiles in large time asymptotics of solutions to damped wave equations." Studia Mathematica 143.2 (2000): 175-197. <http://eudml.org/doc/216815>.

@article{Karch2000,
abstract = {Large time behavior of solutions to the generalized damped wave equation $u_\{tt\} +A u_t +ν B u+F(x,t,u,u_t,∇ u) = 0$ for $(x,t)∈ ℝ^n × [0,∞)$ is studied. First, we consider the linear nonhomogeneous equation, i.e. with F = F(x,t) independent of u. We impose conditions on the operators A and B, on F, as well as on the initial data which lead to the selfsimilar large time asymptotics of solutions. Next, this abstract result is applied to the equation where $Au_t = u_t$, $Bu = -Δu$, and the nonlinear term is either $|u_t|^\{q-1\}u_t$ or $|u|^\{α-1\}u$. In this case, the asymptotic profile of solutions is given by a multiple of the Gauss-Weierstrass kernel. Our method of proof does not require the smallness assumption on the initial conditions.},
author = {Karch, Grzegorz},
journal = {Studia Mathematica},
keywords = {generalized wave equation with damping; the Cauchy problem; large time behavior of solutions; selfsimilar solutions; Cauchy problem; selfsimilar large time asymptotics; Gauss-Weierstrass kernel},
language = {eng},
number = {2},
pages = {175-197},
title = {Selfsimilar profiles in large time asymptotics of solutions to damped wave equations},
url = {http://eudml.org/doc/216815},
volume = {143},
year = {2000},
}

TY - JOUR
AU - Karch, Grzegorz
TI - Selfsimilar profiles in large time asymptotics of solutions to damped wave equations
JO - Studia Mathematica
PY - 2000
VL - 143
IS - 2
SP - 175
EP - 197
AB - Large time behavior of solutions to the generalized damped wave equation $u_{tt} +A u_t +ν B u+F(x,t,u,u_t,∇ u) = 0$ for $(x,t)∈ ℝ^n × [0,∞)$ is studied. First, we consider the linear nonhomogeneous equation, i.e. with F = F(x,t) independent of u. We impose conditions on the operators A and B, on F, as well as on the initial data which lead to the selfsimilar large time asymptotics of solutions. Next, this abstract result is applied to the equation where $Au_t = u_t$, $Bu = -Δu$, and the nonlinear term is either $|u_t|^{q-1}u_t$ or $|u|^{α-1}u$. In this case, the asymptotic profile of solutions is given by a multiple of the Gauss-Weierstrass kernel. Our method of proof does not require the smallness assumption on the initial conditions.
LA - eng
KW - generalized wave equation with damping; the Cauchy problem; large time behavior of solutions; selfsimilar solutions; Cauchy problem; selfsimilar large time asymptotics; Gauss-Weierstrass kernel
UR - http://eudml.org/doc/216815
ER -

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