Geodesic
On an instantaneous blow-up of solutions of evolutionary problems on the half-line
Izvestiya. Mathematics , Tome 82 (2018) no. 5, pp. 914-930.

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We consider some initial-boundary value problems on the half-line for ‘1+1’-dimensional equations of Sobolev type with homogeneous boundary conditions at the beginning of the half-line. We show that weak solutions of these problems are absent even locally in time. Moreover, we consider problems on an interval with the same boundary conditions on one of the ends of the interval $[0,L]$. We prove the local in time (unique) solubility of the problems under consideration in the classical sense, and obtain sufficient conditions for the blow-up of these solutions in finite time. Using the upper bounds thus obtained for the blow-up times for classical solutions of the corresponding problems, we show that the blow-up time tends to zero as $L\to+\infty$. Thus, a classical solution on the line is also absent, even locally, and we describe an algorithm for the subsequent numerical diagnosis of the instantaneous blow-up on the half-line.
Keywords: non-linear equations of Sobolev type, blow-up, local solubility, non-linear capacity, bounds for the blow-up time. 
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M. O. Korpusov. On an instantaneous blow-up of solutions of evolutionary problems on the half-line. Izvestiya. Mathematics , Tome 82 (2018) no. 5, pp. 914-930. http://geodesic.mathdoc.fr/item/IM2_2018_82_5_a2/

[1] H. Brezis, X. Cabré, “Some simple nonlinear PDE's without solutions”, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 1:2 (1998), 223–262 | MR | Zbl

[2] X. Cabré, Y. Martel, “Existence versus explosion instantanée pour des équations de la chaleur linéaires avec potentiel singulier [Existence versus instantaneous blow-up for linear heat equations with singular potentials]”, C. R. Acad. Sci. Paris Sér. I Math., 329:11 (1999), 973–978 | DOI | MR | Zbl

[3] F. B. Weissler, “Local existence and nonexistence for semilinear parabolic equations in $L^p$”, Indiana Univ. Math. J., 29:1 (1980), 79–102 | DOI | MR | Zbl

[4] È. Mitidieri, S. I. Pokhozhaev, “A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities”, Proc. Steklov Inst. Math., 234 (2001), 1–362 | MR | Zbl

[5] V. A. Galaktionov, J.-L. Vázquez, “The problem of blow-up in nonlinear parabolic equations”, Current developments in partial differential equations (Temuco, 1999), Discrete Contin. Dyn. Syst., 8:2 (2002), 399–433 | DOI | MR | Zbl

[6] J. A. Goldstein, I. Kombe, “Instantaneous blow up”, Advances in differential equations and mathematical physics (Birmingham, AL, 2002), Contemp. Math., 327, Amer. Math. Soc., Providence, RI, 2003, 141–150 | DOI | MR | Zbl

[7] Y. Giga, N. Umeda, “On instant blow-up for semilinear heat equations with growing initial data”, Methods Appl. Anal., 15:2 (2008), 185–195 | DOI | MR | Zbl

[8] E. I. Galakhov, “On the absence of local solutions of several evolutionary problems”, Math. Notes, 86:3 (2009), 314–324 | DOI | DOI | MR | Zbl

[9] E. I. Galakhov, “On the instantaneous blow-up of solutions of some quasilinear evolution problems”, Differ. Equ., 46:3 (2010), 329–338 | DOI | MR | Zbl

[10] M. O. Korpusov, “Critical exponents of instantaneous blow-up or local solubility of non-linear equations of Sobolev type”, Izv. Math., 79:5 (2015), 955–1012 | DOI | DOI | MR | Zbl

[11] A. A. Panin, “On local solvability and blow-up of solutions of an abstract nonlinear Volterra integral equation”, Math. Notes, 97:6 (2015), 892–908 | DOI | DOI | MR | Zbl