On critical exponents for some quasilinear parabolic equations

We study the Cauchy problem for the quasilinear parabolic equation where p > 1 is a parameter and ψ is a smooth, bounded function on (1, ∞) with − ⩽ sψ′(s)/ψ(s) ⩽ θ for some θ > 0. If 1 1 + 2/N, there are global, positive solutions for small initial data.

[1]  Masayoshi Tsutsumi,et al.  Existence and Nonexistence of Global Solutions for Nonlinear Parabolic Equations , 1972 .

[2]  G. M. Lieberman.  Hölder continuity of the gradient of solutions of uniformly parabolic equations with conormal boundary conditions , 1987 .

[3]  Peter Meier,et al.  On the critical exponent for reaction-diffusion equations , 1990 .

[4]  Kantaro Hayakawa,et al.  On Nonexistence of Global Solutions of Some Semilinear Parabolic Differential Equations , 1973 .

[5]  Howard A. Levine,et al.  The Role of Critical Exponents in Blowup Theorems , 1990, SIAM Rev..

[6]  O. Ladyženskaja.  Linear and Quasilinear Equations of Parabolic Type , 1968 .

[7]  Hiroshi Tanaka,et al.  On the growing up problem for semilinear heat equations , 1977 .

[8]  K. Hayakawa,et al.  On nonexistence of global solutions of some semilinear parabolic equations , 1973 .

[9]  M. Kwong,et al.  Semilinear elliptic problems in annular domains , 1989 .

[10]  Lawrence E. Payne,et al.  Nonexistence of global weak solutions for classes of nonlinear wave and parabolic equations , 1976 .

[11]  Geometrical implications of the existence of very smooth bump functions in Banach spaces , 1989 .

[12]  Howard A. Levine,et al.  Fujita type results for convective-like reaction diffusion equations in exterior domains , 1989 .

[13]  Otared Kavian,et al.  Remarks on the large time behaviour of a nonlinear diffusion equation , 1987 .

[14]  Howard A. Levine,et al.  The value of the critical exponent for reaction-diffusion equations in cones , 1990 .

[15]  E. Diurkovich.  Numerical determination with given accuracy of the optimal response time of a system , 1983 .

[16]  Fred B. Weissler,et al.  An L∞ blow‐up estimate for a nonlinear heat equation , 1985 .

[17]  Peter Meier,et al.  Blow-up of solutions of semilinear parabolic differential equations , 1988 .