Blow up of solutions of the Cauchy problem for a wave equation with nonlinear damping and source terms and positive initial energy

Authors:
Howard A. Levine and Grozdena Todorova

Journal:
Proc. Amer. Math. Soc. **129** (2001), 793-805

MSC (1991):
Primary 35L15, 35Q72

DOI:
https://doi.org/10.1090/S0002-9939-00-05743-9

Published electronically:
September 19, 2000

MathSciNet review:
1792187

Full-text PDF

Abstract | References | Similar Articles | Additional Information

In this paper we consider the long time behavior of solutions of the initial value problem for semi-linear wave equations of the form

Here

We prove that if then for any there are choices of initial data from the energy space with initial energy such that the solution blows up in finite time. If we replace by , where is a sufficiently slowly decreasing function, an analogous result holds.

**1.**BALL, J., Remarks on blow-up and nonexistence theorems for nonlinear evolution equations,*Quart. J. Math. Oxford*(2)**28**(1977), 473-486. MR**57:13150****2.**GEORGIEV, V. AND G. TODOROVA, Existence of a solution of the wave equation with nonlinear damping and source terms,*J. Diff. Eqs.***109**(1994), 295-308. MR**95b:35141****3.**GLASSEY, R. T., Blow-up theorems for nonlinear wave equations,*Math. Z.***132**(1973), 183-203. MR**49:5549****4.**Ikehata, R., Some remarks on the wave equations with nonlinear damping and source terms*Nonl. Anal. T. M. A*(to appear). MR**97i:35117****5.**KAWARADA, K., On solutions of nonlinear wave equations,*J. Phys. Soc. Jap.*(1971), 280-282.**6.**KELLER, J. B., On solutions of nonlinear wave equations,*Comm. Pure Appl. Math***10**(1957), 523-530. MR**20:3371****7.**KNOPS, R. J., Levine, H. A. and Payne L. E., Nonexistence, instability and growth theorems for solutions to an abstract nonlinear equation with applications to elastodynamics,*Arch. Rational Mech. Anal.***55**(1974), 52-72. MR**51:1093****8.**LEVINE, H.A., Instability and nonexistence of global solutions of nonlinear wave equations of the form ,*Trans. Amer. Math. Soc.***192**(1974), 1-21. MR**49:9436****9.**LEVINE, H.A., Some additional remarks on the nonexistence of global solutions to nonlinear wave equations,*SIAM J. Math. Anal.***5**(1974), 138-146. MR**53:3525****10.**LEVINE, H.A., Nonexistence of global weak solutions to some properly and improperly posed problems of mathematical physics: The method of unbounded Fourier coefficients,*Math. Ann.***214**(1975), 205-220. MR**52:6200****11.**LEVINE, H.A., PARK, S. R. AND SERRIN, J., Global existence and global nonexistence of solutions of the Cauchy problem for a nonlinearly damped wave equation,*JMAA***228**(1998), 181-205. MR**99k:35124****12.**LEVINE, H.A., PUCCI, P. AND SERRIN, J., Some remarks on global nonexistence for nonautonomous abstract evolution equations,*Contemporary Math.***208**(1997), 253-263. MR**98j:34124****13.**LEVINE, H.A. AND SERRIN, J., A global nonexistence theorem for quasilinear evolution equations with dissipation,*Arch. Rational. Mech. Anal.***137**(1997), 341-361. MR**99b:34110****14.**NAKAO, M. AND ONO, K., Existence of global solutions to the Cauchy problem for the semilinear dissipative wave equations,*Math. Z.***214**(1993), 325 - 342. MR**94h:35167****15.**OHTA, M., Remarks on blow up of solutions for nonlinear evolution equations of second order, (to appear). MR**99m:35167****16.**ONO, K. On global solutions and blowup solutions of nonlinear Kirchhoff strings with nonlinear dissipation,*J. Math. Anal. and Appl.***216**(1997), 321-342. MR**99e:35219****17.**PAYNE, L. E. AND SATTINGER, D., Saddle points and instability of nonlinear hyperbolic equations,*Israel Math. J.***22**(1981), 273-303. MR**53:6112****18.**PUCCI, P. AND SERRIN, J. Global nonexistence for abstract evolution equations with positive initial energy,*JDE***109**(1998)). MR**2000a:34119****19.**STRAUGHAN, B., Further global nonexistence theorems for abstract nonlinear wave equations*Proc. AMS***48**(2)(1975) 381-390. MR**51:1518****20.**STRAUSS, W., Nonlinear Wave Equations*CBMS Regional Conference Series in Mathematics, Am. Math. Soc.***73**(1989). MR**91g:35002****21.**STRAUSS, W., The Energy Method in Nonlinear Partial Differential Equations,*Notas De Matematics #47*, Istit. di Mat. Conselho Nacional de Pesquisas, Rio de Janeiro 1969. MR**42:8051****22.**TSUTSUMI, H., On solutions of semilinear differential equations in a Hilbert space,*Math. Japonicea***17**(1972), 173-193.**23.**TODOROVA, G., The Cauchy problem for nonlinear wave equations with nonlinear damping and source terms*J. Nonl. Anal. TMA***41**7-8 (2000), 891-905.**24.**TODOROVA, G., The Cauchy problem for nonlinear wave equations with nonlinear damping and source terms*C. R. Acad. Sci. Paris***326***Série I*(1998), 191-196. MR**99e:35154****25.**TODOROVA, G., Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms*C. R. Acad. Sci. Paris***328***Série I*(1999), 117-122. MR**99j:35144****26.**TODOROVA, G., Stable and unstable sets for the Cauchy problem for a nonlinear wave equation with nonlinear damping and source terms*JMAA***239**(1999), 213-226. CMP**2000:04****27.**VITILLARO, E., Global nonexistence theorems for a class of evolution equations with dissipation*Arch. Rat. Mech. Anal.*,**149**, 2(1999), 155-182. MR**2000:04**

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Additional Information

**Howard A. Levine**

Affiliation:
Department of Mathematics, Iowa State University, Ames, Iowa 50011

Email:
halevine@iastate.edu

**Grozdena Todorova**

Affiliation:
Institute of Mathematics, Bulgarian Academy of Sciences, Boul. Acad. Bonchev bl.8, Sofia 1113, Bulgaria

Address at time of publication:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455

Email:
grozdena@bas.bg, todorova@math.umm.edu

DOI:
https://doi.org/10.1090/S0002-9939-00-05743-9

Received by editor(s):
May 11, 1999

Published electronically:
September 19, 2000

Additional Notes:
The first author was supported in part by NATO grant CRG-95120. The second author was supported by the Institute for Theoretical and Applied Physics at Iowa State University.

Communicated by:
David S. Tartakoff

Article copyright:
© Copyright 2000
American Mathematical Society