zbMATH — the first resource for mathematics

Critical nonlinearities in partial differential equations. (English) Zbl 1205.35024
Summary: This paper focuses on the role of critical nonlinearities within the framework of global solvability of nonlinear PDE’s. In particular, we present some new approach to blow-up issues for nonlinear problems.

35B33 Critical exponents in context of PDEs
35B40 Asymptotic behavior of solutions to PDEs
35B44 Blow-up in context of PDEs
35K55 Nonlinear parabolic equations
PDF BibTeX Cite
Full Text: DOI
[1] Ferreira R., de Pablo A., Vazquez J.L.: Classification of blow-up with nonlinear diffusion and localized reaction. J. Differential Equations 231(N1), 195–211 (2006) · Zbl 1110.35034
[2] Fujita H.: On the blowing up of solutions to the Cauchy problem for u t = {\(\Delta\)}u + u 1+{\(\alpha\)} . J. Fac. Sci. Univ. Tokyo, Sect. 1A Math. 13, 109–124 (1966)
[3] Galaktionov V.A.: On conditions for there to be no global solutions of a class of quasilinear parabolic equations. USSR Comput. Math. and Math. Phys. 22(N2), 73–90 (1982) · Zbl 0548.35068
[4] Galaktionov V.A.: Blow-up for quasilinear heat equations with critical Fujita’s exponents. Proceedings of the Royal Society of Edinburgh, Section A: Mathematics 124(N3), 517–525 (1994) · Zbl 0808.35053
[5] V. A. Galaktionov, S. P. Kurdyumov, A. P. Mikhailov and A. A. Samarskii, Blow-up in quasilinear parabolic equations, de Gruyter Expositions in Mathematics, 19. Walter de Gruyter and Co., Berlin, 1995. xxii+535 pp. · Zbl 1020.35001
[6] Galaktionov V.A., Mitidieri E., Pohozaev S.I.: Capacity induced by a nonlinear operator and applications. Georgian Math. J. 15(N3), 501–516 (2008) · Zbl 1157.35305
[7] Galaktionov V.A., Mitidieri E., Pohozaev S.I.: Existence and nonexistence of global solutions of the Kuramoto-Sivashinsky equation. Dokl. Akad. Nauk 419(N4), 439–442 (2008) · Zbl 1152.35101
[8] Gidas B., Spruck J.: A priori bounds for positive solutions of nonlinear elliptic equations. Comm. Partial Differential Equations 6, 883–901 (1981) · Zbl 0462.35041
[9] Kato T.: Blow-up of solutions of some nonlinear hyperbolic equations. Comm. Pure Appl. Math. 32, 501–505 (1980) · Zbl 0432.35056
[10] Kuramoto Y., Tsuzuki T.: On the formation of dissipative structures in reaction-diffusion systems. Progr. Theoret. Phys. 54, 687–699 (1975)
[11] Levine H.: The role of critical exponents in blowup theorems. SIAM Rev. 32(2), 262–288 (1990) · Zbl 0706.35008
[12] Levine H., Serrin J.: Global nonexistence theorems for quasilinear evolution equations with dissipation. Arch. Rational Mech. Anal. 137(4), 341–361 (1997) · Zbl 0886.35096
[13] Lions J.L.: Quelques méthodes de resolution des problèmes aux limites non linéaires. Dunod; Gauthier-Villars, Paris (1969)
[14] V.G. Maz’ya, Sobolev spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985.
[15] E. Mitidieri and S. I. Pohozaev, A priori estimates and nonexistence of solutions to nonlinear partial differential equations and inequalities, Moscow, Nauka, 2001 (Proc. Steklov Inst. Math., v. 234). · Zbl 1074.35500
[16] Mitidieri E., Pohozaev S.I.: Towards a unified approach to nonexistence of solutions for a class of differential inequalities. Milan J. Math. 72, 129–162 (2004) · Zbl 1115.35157
[17] Mitidieri E., Pohozaev S.I.: Liouville Theorems for Some Classes of Nonlinear Nonlocal Problems. Proc. Steklov Math. Inst. 248, 164–184 (2005)
[18] Pohozaev S.I.: On eigenfunctions of {\(\Delta\)}u + {\(\lambda\)}f(u) = 0. Doklady Acad. Sci. SU 165(1), 36–39 (1965)
[19] Pohozaev S.I.: On eigenfunctions of quasilinear elliptic problems. Matemat. Sbornik 82(N 2), 192–212 (1970)
[20] Pohozaev S.I.: On equations of the form {\(\Delta\)}u = f(x, u, Du). Matemat. Sbornik 113(N 2), 324–338 (1980) · Zbl 0457.35032
[21] Pohozaev S.I.: On quasilinear elliptic equations of higher order. Differentsialnyje Uravnenija 17(N 1), 115–128 (1981)
[22] Pohozaev S.I.: On solvability of quasilinear elliptic equations of arbitrary order. Matemat. Sbornik. 117(N 2), 251–265 (1982) · Zbl 0491.35020
[23] Pohozaev S.I.: On a priori estimates for solutions of quasilinear elliptic equations of arbitrary order. Differentsialnyje Uravnenija 19(N 1), 101–110 (1983)
[24] Pohozaev S.I.: Essential nonlinear capacity induced by differential operators. Doklady Russ. Acad. Sci. 357(N5), 352–354 (1997)
[25] Pohozaev S.I.: The General Blow-up Theory for Nonlinear PDE’s, in ”Function Spaces, Differential Operators and Nonlinear Analysis: The Hans Triebel Anniversary Volume”, pp. 141–159. Birkhauser, Basel (2003) · Zbl 1064.35219
[26] Pohozaev S.I.: On multidimensional scalar conservation laws. Mat. Sb. 194(N1), 147–160 (2003)
[27] Pohozaev S.I.: On the nonexistence of global solutions of the Hamilton-Jacobi equation. Differentsialnyje Uravnenija 44(N10), 1467–1477 (2008) · Zbl 1194.35103
[28] Pohozaev S.I.: On the blow-up for Kuramoto-Sivashinskii equation. Mat. Sb. 199(N9), 97–106 (2008)
[29] Pohozaev S.I.: On the global solvability of the Kuramoto–Sivashinsky equation under bounded initial data. Matem. Sb. 200(N7), 131–144 (2009)
[30] Sivashinsky G.I.: Nonlinear analysis of hydrodynamic instability in laminar flames. Acta Astronaut. 4, 1177–1206 (1977) · Zbl 0427.76047
[31] J. L. Vazquez, The problems of blow-up for nonlinear heat equations. Complete blow-up and avalanche formation, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., v. 15, N 3-4 (2004), 281–300.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.