Collapse of the solutions of parabolic and hyperbolic equations with nonlinear boundary conditions
It is shown that the solutions of linear and quasilinear equations of parabolic and hyperbolic type may collapse because of the presence of nonlinearities in the boundary conditions.
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- 1.O. A. Ladyzhenskaya (Ladyzenskaja), V. A. Solonnikov, and N. N. Ural'tseva (Ural'ceva), Linear and Quasilinear Equations of Parabolic Type, Amer. Math. Soc., Providence (1968).Google Scholar
- 2.H. A. Levine, “Instability and nonexistence of global solutions to nonlinear wave equations of the form Putt =−Au +F(u),” Trans. Am. Math. Soc.,192, 1–21 (1974).Google Scholar
- 3.V. K. Kalantarov and O. A. Ladyzhenskaya, “The occurrence of collapse for quasilinear equations of parabolic and hyperbolic types,” Zap. Nauchn. Sem. Leningr. Otd. Mat. Inst.,69, 77–102 (1977).Google Scholar
- 4.S. I. Pokhozhaev, “Questions of the absence of solutions of nonlinear boundary-value problems,” in: Proc. of an All-Union Conf. on Partial Differential Equations [in Russian], Moscow State Univ. (1978), pp. 200–203.Google Scholar
- 5.F. John, “Blow-up of solutions of nonlinear wave equations in three space dimensions,” Manuscr. Math.,28, 235–268 (1979).Google Scholar
- 6.V. A. Galaktionov, “On conditions for the absence of global solutions of a certain class of quasilinear parabolic equations,” Zh. Vychisl. Mat. Mat. Fiz.,22, No. 2, 322–338 (1982).Google Scholar
- 7.H. A. Levine and L. E. Payne, “Some nonexistence theorems for initial-boundary-value problems with nonlinear boundary constraints,” Proc. Am. Math. Soc.,46, 277–284 (1974).Google Scholar
- 8.H. A. Levine and L. E. Payne, “Nonexistence theorems for the heat equation with non-linear boundary conditions and for the porous medium equation backward in time,” J. Diff. Equations,16, 319–334 (1974).Google Scholar
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