Existence and nonexistence of global solutions of the wave equation with a nonlinear boundary condition
Authors:
Azmy S. Ackleh and Keng Deng
Journal:
Quart. Appl. Math. 59 (2001), 153-158
MSC:
Primary 35L05; Secondary 35B40, 35L15, 35L20
DOI:
https://doi.org/10.1090/qam/1811100
MathSciNet review:
MR1811100
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Abstract: We study the initial-boundary value problem \[ {u_{tt}} = {u_{xx}}, \qquad 0 < x < \infty , \qquad t > 0,\] \[ - {u_x}\left ( 0, t \right ) = h\left ( u\left ( 0, t \right ) \right ), \qquad t > 0,\] \[ u\left ( x, 0 \right ) = f\left ( x \right ), \qquad {u_t}\left ( x, 0 \right ) = g\left ( x \right ), \qquad 0 < x < \infty .\] We establish criteria for existence and nonexistence of global solutions, and we present the growth rate at blow-up.
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N. Bleistein and R. A. Handelsman, Asymptotic Expansion of Integrals, Holt, Rinehart and Winston, New York, 1975
C. Corduneanu, Integral Equations and Applications, Cambridge University Press, New York, 1991
V. A. Galaktionov and H. A. Levine, On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary, Israel J. Math. 94, 125–146 (1996)
R. T. Glassey, Finite-time blow-up for solutions of nonlinear wave equations, Math. Z. 177, 323–340 (1981)
T. Kato, Blow-up of solutions of some nonlinear hyperbolic equations, Comm. Pure Appl. Math. 32, 501–505 (1980)
W. E. Olmstead and C. A. Roberts, Explosion in a diffusive strip due to a concentrated nonlinear source, Methods Math. Anal. 1, 434–445 (1994)
C. A. Roberts and W. E. Olmstead, Growth rates for blow-up solutions of nonlinear Volterra equations, Quart. Appl. Math. 54, 153–160 (1996)
T. Sideris, Nonexistence of global solutions to semilinear wave equations in high dimensions, J. Differential Equations 52, 378–406 (1984)
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Article copyright:
© Copyright 2001
American Mathematical Society
