Non-homogeneous nonlinear damped wave equations in unbounded domains. (English) Zbl 0728.35071

Racke, Reinhard

Math. Methods Appl. Sci. 13, No. 6, 481-491 (1990).

Author’s abstract: The author presents a global existence theorem of solutions for the following Cauchy problem: \[ u_{tt}-\partial_ ia_{ik}\partial_ ku+u_ t=f(t,x,u,u_ t,\nabla u,\nabla u_ t,\nabla^ 2u)\text{ in } \{t>0,\quad x\in \Omega \}, \]
\[ u(0,x)=u^ 0(x),\quad u_ t(0,x)=u^ 1(x)\text{ on } \{t=0,\quad x\in \Omega \}. \] The domain \(\Omega\) equals \(R^ 3\) or \(\Omega\) is an exterior domain in \(R^ 3\) with smoothly bounded star-shaped complement. In the latter case the boundary condition is of Dirichlet type, i.e., \(u|_{\partial \Omega}=0\). The main theorem is obtained for small data \(\{u^ 0,u^ 1\}\) under certain conditions on the coefficients \(a_{ik}\). The \(L^ p-L^ q\) decay rates of solutions of the linearized problem, based on a previously introduced generalized eigenfunction expansion ansatz, are used to derive the necessary a priori estimates.

Reviewer: M.Tsuji (Kyoto)

MSC:

35L70	Second-order nonlinear hyperbolic equations
35B40	Asymptotic behavior of solutions to PDEs
35B45	A priori estimates in context of PDEs
35L20	Initial-boundary value problems for second-order hyperbolic equations

Keywords:

nonlinear damped wave equations; global existence; small data; decay rates

References:

[1]	Sobolev Spaces, Academic Press, New York, 1975. · Zbl 0314.46030
[2]	Lectures on Elliptic Boundary Value Problems, Van Nostrand, Princeton, 1965.
[3]	Bloom, J. Math. Anal. Appl. 44 pp 310– (1973)
[4]	Klainerman, Comm. Pure Appl. Math. 33 pp 43– (1980)
[5]	Klainerman, Comm. Pure Appl. Math. 36 pp 133– (1983)
[6]	Li, Comm. Part. Diff. Equ. 13 pp 383– (1988)
[7]	Matsumura, Publ. RIMS, Kyoto Univ. 13 pp 349– (1977)
[8]	Morawetz, Comm. Pure Appl. Math. 21 pp 187– (1968)
[9]	Ponce, Nonlinear Anal. T.M.A. 9 pp 399– (1985)
[10]	Racke, J. reine angew. Math.
[11]	Racke, Asymptotic Analysis 3 pp 105– (1990)
[12]	Racke, Nonlinear Anal. T.M.A.
[13]	Shatah, J. Diff. Equations. 46 pp 409– (1982)
[14]	Shibata, Fune. Ekra. 25 pp 303– (1982)
[15]	Shibata, Tsukuba J. Math. 7 pp 1– (1983)
[16]	Shibata, Math. Z. 191 pp 165– (1986)
[17]	Shibata, Nonlinear Anal. T.M.A. 11 pp 335– (1987)