Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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QUALITATIVE PROPERTIES OF WEAK SOLUTIONS FOR p-LAPLACIAN EQUATIONS WITH NONLOCAL SOURCE AND GRADIENT ABSORPTION

  • Chaouai, Zakariya (Center of Mathematical Research of Rabat (CeReMAR) Laboratory of Mathematical Analysis and Applications (LAMA) Faculty of Sciences Mohammed V University) ;
  • El Hachimi, Abderrahmane (Center of Mathematical Research of Rabat (CeReMAR) Laboratory of Mathematical Analysis and Applications (LAMA) Faculty of Sciences Mohammed V University)
  • Received : 2019.08.01
  • Accepted : 2020.03.26
  • Published : 2020.07.31
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Abstract

We consider the following Dirichlet initial boundary value problem with a gradient absorption and a nonlocal source $$\frac{{\partial}u}{{\partial}t}-div({\mid}{\nabla}u{\mid}^{p-2}{\nabla}u)={\lambda}u^k{\displaystyle\smashmargin{2}{\int\nolimits_{\Omega}}}u^sdx-{\mu}u^l{\mid}{\nabla}u{\mid}^q$$ in a bounded domain Ω ⊂ ℝN, where p > 1, the parameters k, s, l, q, λ > 0 and µ ≥ 0. Firstly, we establish local existence for weak solutions; the aim of this part is to prove a crucial priori estimate on |∇u|. Then, we give appropriate conditions in order to have existence and uniqueness or nonexistence of a global solution in time. Finally, depending on the choices of the initial data, ranges of the coefficients and exponents and measure of the domain, we show that the non-negative global weak solution, when it exists, must extinct after a finite time.

Keywords

Acknowledgement

The authors would like to thank the anonymous referee for the detailed comments and suggestions which have been helpful to further improve the paper.

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