Global existence and nonexistence of solutions for Newtonian filtration equation

Abstract

In this paper, we study the initial boundary value problem of Newtonian filtration equation in a cone-like do main , where , , with . Let denote the smallest Dirichlet eigenvalue for the Laplace–Beltrami operator on and let l denote the positive root of . We prove that if , then the problem has no global nonnegative solutions for any nonnegative unless ; if , then the problem has global solutions for some .

Keywords:

• Newtonian filtration equation
• nonexistence
• existence

AMS Subject Classifications:

• 35B33
• 35K55
• 35K65

Acknowledgements

The authors would like to express their deep thanks to the referee’s valuable suggestions for the revision and improvement of the manuscript.

Notes

No potential conflict of interest was reported by the authors.