Geodesic
Selfsimilar profiles in large time asymptotics of solutions to damped wave equations
Studia Mathematica, Tome 143 (2000) no. 2, pp. 175-197

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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.
DOI : 10.4064/sm-143-2-175-197
Keywords: generalized wave equation with damping, the Cauchy problem, large time behavior of solutions, selfsimilar solutions

Grzegorz Karch 1

1 
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Grzegorz Karch. Selfsimilar profiles in large time asymptotics of solutions to damped wave equations. Studia Mathematica, Tome 143 (2000) no. 2, pp. 175-197. doi: 10.4064/sm-143-2-175-197

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