Abstract
In this paper, we prove the existence of Fujita-type critical exponents for x-dependent fully nonlinear uniformly parabolic equations of the type
These exponents, which we denote by p(F), determine two intervals for the p values: in ]1,p(F)[, the positive solutions have finite-time blow-up, and in ]p(F), +∞[, global solutions exist. The exponent p(F) = 1 + 1/α(F) is characterized by the long-time behavior of the solutions of the equation without reaction terms
When F is a x-independent operator and p is the critical exponent, that is, p = p(F). We prove as main result of this paper that any non-negative solution to (*) has finite-time blow-up. With this more delicate critical situation together with the results of Meneses and Quaas (J Math Anal Appl 376:514–527, 2011), we completely extend the classical result for the semi-linear problem.
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Meneses, R., Quaas, A. Existence and non-existence of global solutions for uniformly parabolic equations. J. Evol. Equ. 12, 943–955 (2012). https://doi.org/10.1007/s00028-012-0162-2
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DOI: https://doi.org/10.1007/s00028-012-0162-2