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Blow-up for nonlinear wave equations with slowly decaying data. (English) Zbl 0814.35077

We consider classical solutions to the initial value problem for the “accretive” equation \(\square u = | u_ t |^ p\) in \(\mathbb{R}^ n \times [0, \infty)\) with “small” data. If the data have compact support, it is partially known that there is a number \(p_ 0(n)\) such that solutions blow-up in finite time for \(1< p \leq p_ 0 (n)\) and a global solution exists for \(p > p_ 0(n)\). In this paper, we shall show for \(n = 4,5\) that, if the support of data is noncompact, there are blowing-up solutions even for \(p > p_ 0 (n)\) because of the “bad” spatial decay of the initial data.

MSC:

35L70 Second-order nonlinear hyperbolic equations
35B40 Asymptotic behavior of solutions to PDEs
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References:

[1] Agemi, R.: Blow-up of solutions to nonlinear wave equations in two space dimensions. Manuscripta Math.73, 153–162 (1991) · Zbl 0763.35060
[2] Asakura, F.: Existence of spherically symmetric global solution to the semi-linear wave equationu u u=au t 2 +b()2 in five space dimensions J. Math. Kyoto Univ24, 361–380 (1984) · Zbl 0562.35058
[3] Courant, R., Hilbert, D.: Methods of Mathematical Physics II. New York: Interscience 1962 · Zbl 0099.29504
[4] Glassey, R.T.: Blow-up theorems for nonlinear wave equations. Math. Z.132, 183–203 (1973) · Zbl 0254.35078
[5] John, E.: Plane waves and spherical means applied to partial differential equations. New York: Interscience 1955 · Zbl 0067.32101
[6] John, F.: Blow-up for quasi-linear wave equations in three space dimensions. Comm. Pure Appl. Math.34, 29–51 (1981) · Zbl 0453.35060
[7] John, F.: Non-existence of global solutions of \(\square u = \tfrac{\partial }{{\partial t}}F(u_t )\) in two or three space dimensions. Rend. Circ. Mat. Palermo (2) Suppl.8, 229–249 (1985) · Zbl 0649.35060
[8] Klainerman, S.: Uniform decay estimates and the Lorentz invariance of the classical wave equation. Comm. Pure Appl. Math.38, 321–332 (1985) · Zbl 0635.35059
[9] Klainerman, S.: Remarks on the global Sobolev inequalities in Minkowski space \(\mathbb{R}\) n . Comm. Pure Appl. Math.40, 111–117 (1987) · Zbl 0686.46019
[10] Kubo, H.: Blow-up for semilinear wave equations with initial data of slow decay in low space dimensions. Differential Integral Equations7, 315–321 (1994) · Zbl 0818.35067
[11] Kubo, H.: Asymptotic behaviors of solutions to semilinear wave equations with initial data of slow decay. Hokkaido Univ. Preprint Ser. Math. #196 (1993) (to appear) Math. Meth. Appl. Sci.
[12] Masuda, K.: Blow-up of solutions for quasi-linear wave equations in two space dimensions. Lecture Notes in Num. Appl. Anal.6, 87–91 (1983)
[13] Rammaha, M.A.: Finite-time blow-up for nonlinear wave equations in high dimensions. Comm. Partial Differential Equations.12(6), 677–700 (1987) · Zbl 0631.35060
[14] Schaeffer, J.: Wave equation with positive nonlinearities. Ph. D. Thesis, Indiana University (1983)
[15] Schaeffer, J.: Finite-time blow-up foru tt -{\(\Delta\)}u=H(u r ,u t ) in two space dimensions. Comm. Partial Differential Equations.11(5), 513–543 (1986) · Zbl 0592.35088
[16] Sideris, T.C.: Global behavior of solutions to nonlinear wave equations in three space dimensions. Comm. Partial Differential Equations.8(12), 1291–1323 (1983) · Zbl 0534.35069
[17] Takamura, H.: Global existence for nonlinear wave equations with small data of noncompact support in three space dimensions. Comm. Partial Differential Equations.17(1&2), 189–204 (1992) · Zbl 0757.35046
[18] Takamura, H.: Blow-up for semilinear wave equations in four of five space dimensions. Mathematical Research Note 93-001, University of Tsukuba (to appear in Nonlinear Anal.) · Zbl 0814.35078
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