Critical exponents in a degenerate parabolic equation with weighted source

Abstract

This article deals with the Fujita-type theorems to the Cauchy problem of degenerate parabolic equation not in divergence form with weighted source u _t  = u ^p Δu + a(x)u ^q in ℝ ⁿ  × (0, T), where p ≥ 1, q > 1, and the positive weight function a(x) is of the order |x| ^m with m > −2. It was known that for the degenerate diffusion equation in divergence form, the weight function affects both of the critical Fujita exponent and the second critical exponent (describing the critical smallness of initial data required by global solutions via the decay rates of the initial data at space-infinity). Contrarily, it is interesting to prove that the weight function in the present model with degenerate diffusion not in divergence form influences the second critical exponent only, without changing the critical Fujita exponent.

Keywords:

• degenerate parabolic equation
• nonlinear diffusion not in divergence form
• critical exponent
• second critical exponent
• global existence
• blow-up

AMS Subject Classifications:

• Primary 35K55
• 35B33

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant no. 11171048).