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## Studia Mathematica

2000 | 143 | 2 | 175-197
Tytuł artykułu

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

Autorzy
Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
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.
Słowa kluczowe
EN
Czasopismo
Rocznik
Tom
Numer
Strony
175-197
Opis fizyczny
Daty
wydano
2000
otrzymano
2000-09-18
Twórcy
autor
• Institute of Mathematics, Wrocław University, Pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Bibliografia
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• [3] P. Biler, G. Karch and W. A. Woyczy/nski, Asymptotics for multifractal conservation laws, Studia Math. 135 (1999), 231-252.
• [4] P. Biler, G. Karch and W. A. Woyczy/nski, Critical nonlinearity exponent and self-similar asymptotics for Lévy conservation laws, preprint, 1999, 29 pp., http://www.math.uni.wroc.pl/~karch
• [5] D. B. Dix, The dissipation of nonlinear dispersive waves: The case of asymptotically weak nonlinearity, Comm. Partial Differential Equations 17 (1992), 1665-1693.
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• [16] G. Karch, Selfsimilar large time behavior of solutions to Korteweg-de Vries-Burgers equation, Nonlinear Anal. 35 (1999), 199-219.
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• [21] K. Nishihara, Decay properties of solutions of some quasilinear hyperbolic equations with strong damping, Nonlinear Anal. 21 (1993), 17-21.
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• [23] K. Nishihara, Asymptotic behavior of solutions of quasilinear hyperbolic equations with linear damping, ibid. 137 (1997), 384-395.
• [24] K. Nishihara and T. Yang, Boundary effect on asymptotic behaviour of solutions to the p-system with linear damping, ibid. 156 (1999), 439-458.
• [25] K. Ono, The time decay to the Cauchy problem for semilinear dissipative wave equations, Adv. Math. Sci. Appl. 9 (1999), 243-262.
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Bibliografia
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