Global existence and asymptotic behavior of small solutions for semilinear dissipative wave equations

By K. Ono. 2003

58 citations recorded by the source corpus.

We study the global existence and asymptotic behavior of solutions to the Cauchy problem for the semilinear dissipative wave equations: $\square u + \partial_t u = |u|^{\alpha+1}$, $u|_{t=0}=\varepsilon u_0 \in H^1 \cap L^1$, $\partial_t u |_{t=0} = \varepsilon u_1 \in L^2 \cap L^1$ with a small parameter $\varepsilon>0$. When $N\le 3$ and $2/N<\alpha \le 2/[N-2]^+$, we show the global solvability and derive the sharp rates of the solutions.

Subfields: Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/5cccdaac385928776b3180d0fa2e8ebe91fd7eb0

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In Mathematical Physics

Citations in the corpus

Global asymptotics of solutions to the Cauchy problem for the damped wave equation with absorption (2006, background): “There have been many studies of the non-absorbing semilinear term problem [7,8,13–17,22, 24,31,32,34], etc.”
Damped wave equation with super critical nonlinearities (2004, background): “Later these requirements on the initial data were relaxed in [10] as follows u0 ∈ L(1), ∂u0 ∈ L(2), |α| ≤ 1, u1 ∈ L(1) ∩ L(2), under the additional assumptions on ρ and q such that ρ ≤ 5, q ≤ 6 for the space dimension n = 3 and q < ∞ for the two dimensional case n = 2.”
Decay properties of solutions to the Cauchy problem for the damped wave equation with absorption (2006, background): “See also Ikehata et al. [18], Ikehata and Tanizawa [17], Ono [29,30], Gallay and Raugel [7,8], Karch [20], and references therein for the global existence and its proﬁle.”