Abstract
We consider the problem of nonexistence (blow-up) of solutions of nonlinear evolution equations in the case of a bounded (with respect to the space variables) domain. Following the method of nonlinear capacity based on the application of test functions that are optimal (“characteristic”) for the corresponding nonlinear operators, we obtain conditions for the blowup of solutions to nonlinear initial-boundary value problems. We also show by examples that these conditions are sharp in the class of problems under consideration.
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H. Bellout, S. Benachour, and E. S. Titi, “Finite-Time Singularity versus Global Regularity for Hyper-viscous Hamilton-Jacobi-like Equations,” Nonlinearity 16(6), 1967–1989 (2003).
V. A. Galaktionov and S. I. Pohozaev, “Blowup for Nonlinear Initial-Boundary Value Problems,” Dokl. Akad. Nauk 412(4), 444–447 (2007) [Dokl. Math. 75 (1), 76–79 (2007)].
E. Mitidieri and S. I. Pohozaev, A Priori Estimates and Blow-up of Solutions to Nonlinear Partial Differential Equations and Inequalities (Nauka, Moscow, 2001), Tr. Mat. Inst. im. V.A. Steklova, Ross. Akad. Nauk 234 [Proc. Steklov Inst. Math. 234 (2001)].
S. I. Pohozaev, “Essentially Nonlinear Capacities Induced by Differential Operators,” Dokl. Akad. Nauk 357(5), 592–594 (1997) [Dokl. Math. 56, 924–926 (1997)].
S. I. Pokhozhaev, “Normal Solubility of Nonlinear Equations in Uniformly Convex Banach Spaces,” Funkts. Anal. Prilozh. 3(2), 80–84 (1969) [Funct. Anal. Appl. 3, 147–151 (1969)].
S. I. Pohozaev, “On Weakly Nonlinear Hyperbolic Systems,” Tr. Mosk. Mat. O-va 23, 161–176 (1970) [Trans. Moscow Math. Soc. 23, 60–74 (1972)].
A. G. Sveshnikov, A. B. Al’shin, M. O. Korpusov, and Yu. D. Pletner, Linear and Nonlinear Equations of Sobolev Type (Fizmatlit, Moscow, 2007) [in Russian].
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Original Russian Text © S.I. Pohozaev, 2008, published in Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2008, Vol. 260, pp. 213–226.
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Pohozaev, S.I. On the blow-up of solutions to nonlinear initial-boundary value problems. Proc. Steklov Inst. Math. 260, 204–217 (2008). https://doi.org/10.1134/S008154380801015X
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DOI: https://doi.org/10.1134/S008154380801015X