Non well posedness of parabolic equations with supercritical nonlinearities

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World Scientific Publ. Co. Pte. Ltd.
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In this paper we show that several known critical exponents for nonlinear parabolic problems axe optimal in the sense that supercritical problems are ill posed in a strong sense. We also give an answer to an open problem proposed by Brezis and Cazenave in [9], concerning the behavior of the existence time for critical problems. Our results cover nonlinear heat equations including the case of nonlinear boundary conditions and weigthed spaces settings. In the latter case we show that in some cases the critical exponent is equal to one.
F. Abergel, Existence and finite dimensionality of the global attractor for evolution equations on unbounded domains, J. Differential Equations 83 (1990) 85–108. H. Amann. Nonhomogeneous linear and quasilinear elliptic and parabolic boundary value problems, in Schmeisser/Triebel: Function Spaces, Differential Operators and Nonlinear Analysis, Teubner Texte zur Mathematik, Vol. 133 (1993), pp. 9–126. J. M. Arrieta and A. N. Carvalho, Abstract parabolic problems with critical nonlinearities and applications to Navier-Stokes and heat equations, Trans. Amer. Math. Soc. 352 (2000) 285–310. J. M. Arrieta, A. Carvalho and A. Rodríguez-Bernal, Parabolic problems with nonlinear boundary conditions and critical nonlinearities, J. Differential Equations 165 (1999) 376–406. J. M. Arrieta and A. Rodríguez-Bernal, Localization near the boundary of blowup for reaction-diffusion equations, Dept. Matematica Aplicada U. Complutense, Preprint Series MA-UCM 2003–3, to appear in Comm. Partial Differential Equations. cf. A. V. Babin and M. I. Vishik, Attractors of partial differential evolution equations in unbounded domain, Proc. Roy. Soc. Edinburgh 116A (1990) 221–243. P. Baras, Non-unicité des solutions d'une equation d'évolution non-linéaire, Ann. Fac. Sci. Toulouse Math. 5 (1983) 287–302. P. Baras and L. Cohen, Complete blow up after T max for the solution of a semilinear heat equation, J. Funct. Anal. 71 (1987) 142–174. H. Brezis and T. Cazenáve, A nonlinear heat equation with singular initial data, J. Anal. Math. 68 (1996) 277–304. H. Brezis and A. Friedman, Nonlinear parabolic equations involving measures as initial conditions, J. Math. Pures Appl. 62 (1983) 73–97. M. A. Efendiev and S. V. Zelik, The attractor for a nonlinear reaction-diffusion system in an unbounded domain, Comm. Pure Appl. Math. 54 (2001) 625–688. M. Fila, J. Filo and G. M. Lieberman, Blow up on the boundary for the heat equation, Calc. Var. Partial Differential Equations 10 (2000) 85–99. M. Fila, H. Matano and P. Polčik, Existence of L 1 -connections between equilibria of a semilinear parabolic equation, J. Dynam. Differential Equations 14 (2002) 463–491. M. Fila, P. Souplet and F. Weissler, Linear and nonlinear heat equations in L q δ spaces and universal bounds for global solutions, Math. Ann. 320 (2001) 87–113. A. Friedman and B. McLeod, Blow-up of positive solutions of semilinear heat equations, Indiana Univ. Math. J. 34 (1985) 425–447. V. Galaktionov and J. L. Vazquez, Continuation of blow up solutions of nonlinear heat equations in several space dimensions, Comm. Pure Appl. Math. 50 (1997) 1–67. V. Galaktionov and J. L. Vazquez, The problem of blow-up in nonlinear parabolic equations, Discrete Contin. Dyn. Syst. 8(2) (2002) 399–433. D. Henry, Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840 (Springer-Verlag, Berlin, 1981). A. Haraux and F. Weissler, Nonuniqueness for a semilinear initial value problem, Indiana Univ. Math. J. 31 (1982) 167–189. S. Kaplan, On the growth of solutions of quasi-linear parabolic equations, Comm. Pure Appl. Math. 16 (1963) 305–330. W. M. Ni and P. Sacks, Singular behavior in nonlinear parabolic equations, Trans. Amer. Math. Soc. 287 (1985) 657–671. A. Rodríguez-Bernal, Attractors parabolic problems with nonlinear boundary conditions, critical exponents and singular initial data, J. Differential Equations 181 (2002) 165–196. MR1900464 A. Rodríguez-Bernal and A. Tajdine, Nonlinear balance for reaction-diffusion equations under nonlinear boundary conditions: dissipativity and blow-up, J. Differential Equations 169 (2001) 332–372. H. Tribel, Interpolation Theory, Function Spaces, Differential Operators (North Holland, 1978). W. Walter, On the existence and nonexistence in the large of solution of parabolic differential equations with a nonlinear boundary conditions, SIAM J. Math. Anal. 6(1) (1975) 85–90. F. B. Weissler, Semilinear evolution equations in Banach spaces, J. Funct. Anal. 32 (1979) 277–296. F. B. Weissler, Local existence and nonexistence for semilinear parabolic equations in L p , Indiana Univ. Math. J. 29 (1980) 79–102.