On potential wells and vacuum isolating of solutions for semilinear wave equations. (English) Zbl 1024.35078

The author studies the initial boundary value problem of semilinear wave equation \(u_{tt}-\triangle u=|u|^{p-1}u\) on a bounded domain. He introduces a family of potential wells which include the known potential well as a special case, to prove existence and nonexistence theorems of global solutions for the problem.


35L70 Second-order nonlinear hyperbolic equations
35L20 Initial-boundary value problems for second-order hyperbolic equations
35B45 A priori estimates in context of PDEs
Full Text: DOI


[1] Ball, J. M., Remarks on blow-up and nonexistence theorems for nonlinear evolution equations, Quart. J. Math., 28, 473-486 (1977) · Zbl 0377.35037
[2] Glassay, R. T., Blow-up theorems for nonlinear wave equations, Math. Z., 32, 183-203 (1973) · Zbl 0247.35083
[3] Ikehata, R., Some remarks on the wave equations with nonlinear damping and source terms, Nonlinear Anal., 27, 1165-1175 (1996) · Zbl 0866.35071
[4] Levine, H. A., Instability and nonexistence of global solutions to nonlinear wave equations of the form Pu=\(Au}+F(u)\), Trans. Math. AMS, 92, 1-21 (1974) · Zbl 0288.35003
[5] Levine, H. A., Some additional remarks on the nonexistence of global solutions to nonlinear wave equations, SIAM J. Math. Anal., 5, 138-146 (1974) · Zbl 0243.35069
[6] Lions, J. L., Quelques methods de resolution des problem aux limits nonlinears (1969), Dunod: Dunod Paris · Zbl 0189.40603
[7] Payne, L. E.; Sattinger, D. H., Sadle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22, 273-303 (1975) · Zbl 0317.35059
[8] Sattinger, D. H., On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30, 148-172 (1968) · Zbl 0159.39102
[9] Tsutsumi, M., On solutions of semilinear differential equations in a Hilbert space, Math. Japan, 17, 173-193 (1972) · Zbl 0273.34044
[10] Tsutsumi, M., Existence and nonexistence of global solutions for nonlinear parabolic equations, Publ. RTMS, 8, 211-229 (1972/73) · Zbl 0248.35074
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.