Further global nonexistence theorems for abstract nonlinear wave equations

Author:
Brian Straughan

Journal:
Proc. Amer. Math. Soc. **48** (1975), 381-390

MSC:
Primary 47H15; Secondary 35R20

MathSciNet review:
0365265

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Abstract: It is shown that solutions to a nonlinear abstract wave equation (see equations (2.1)) under specified initial data cannot exist for all time. Precise estimates for the growth of solutions and the 'escape time' are given. A similar nonexistence theorem is also proved for a nonlinear wave equation with dissipation (see equations (3.1)).

**[1]**Robert T. Glassey,*Blow-up theorems for nonlinear wave equations*, Math. Z.**132**(1973), 183–203. MR**0340799****[2]**R. J. Knops, H. A. Levine, and L. E. Payne,*Non-existence, instability, and growth theorems for solutions of a class of abstract nonlinear equations with applications to nonlinear elastodynamics*, Arch. Rational Mech. Anal.**55**(1974), 52–72. MR**0364839****[3]**Howard A. Levine,*Instability and nonexistence of global solutions to nonlinear wave equations of the form 𝑃𝑢_{𝑡𝑡}=-𝐴𝑢+\cal𝐹(𝑢)*, Trans. Amer. Math. Soc.**192**(1974), 1–21. MR**0344697**, 10.1090/S0002-9947-1974-0344697-2**[4]**Howard A. Levine,*Some additional remarks on the nonexistence of global solutions to nonlinear wave equations*, SIAM J. Math. Anal.**5**(1974), 138–146. MR**0399682****[5]**Howard A. Levine,*A note on a nonexistence theorem for nonlinear wave equations*, SIAM J. Math. Anal.**5**(1974), 644–648. MR**0361960****[6]**-,*On the nonexistence of global solutions to a nonlinear Euler-Poisson-Darboux equation*, J. Math. Anal. Appl. (in print).**[7]**Masayoshi Tsutsumi,*On solutions of semilinear differential equations in a Hilbert space*, Math. Japon.**17**(1972), 173–193. MR**0355247**

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DOI:
https://doi.org/10.1090/S0002-9939-1975-0365265-9

Article copyright:
© Copyright 1975
American Mathematical Society