Cauchy problem for degenerate and uniformly parabolic equations with nonlocal source

Abstract

The Cauchy problem of the following equation

is considered, where , , , and . We are initially interested in the problem of finding sharp conditions ensuring the local existence of solutions, and give the proof of the local existence of solutions. Then our attention is focus on the global existence and nonexistence of solutions. In particular, a Fujita’s type critical exponent is obtained. These extend several classical conditions and results to the problem considered here.

Keywords:

• Degenerate and uniformly parabolic equation
• Cauchy problem
• nonlocal source
• critical exponent
• existence and nonexistence of solutions

AMS Subject Classifications:

• 35K15
• 35K55
• 35B33

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

This work was supported by National Natural Science Foundation of China [grant number 11201124], [grant number 11301301]; Outstanding Youth Foundation at Henan Polytechnic University [grant number J2014-03]; the Natural Science Foundation of Shandong Province [grant number BS2013SF027]; Foundation for University Key Teacher by the Henan Province [grant number 2015GGJS-070].