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The role of critical exponents in blow-up theorems: The sequel. (English) Zbl 0942.35025
Consider the Cauchy problem in N for the equation u t =Δu+u p , where p>1 and u0. In 1966, H. Fujita [J. Fac. Sci., Univ. Tokyo, Sect. I 13, 109-124 (1966; Zbl 0163.34002)] showed that this problem does not have global nontrivial solutions if p<p c :=1+2/N whereas both global and non-global positive solutions exist if p>p c . The exponent p c is called Fujita’s critical exponent. The authors discuss various Fujita-type results which have appeared in the literature since 1990. These results include degenerate equations, problems in unbounded domains and on manifolds, problems with inhomogeneous boundary conditions, cooperative systems of equations. Moreover, the paper contains a section with open problems.

MSC:
35B33Critical exponents (PDE)
35K55Nonlinear parabolic equations
35B35Stability of solutions of PDE
35B40Asymptotic behavior of solutions of PDE