Results of nonexistence of solutions for some nonlinear evolution problems

Medjahed Djilali; Ali Hakem

Commentationes Mathematicae Universitatis Carolinae (2019)

  • Volume: 60, Issue: 2, page 269-284
  • ISSN: 0010-2628

Abstract

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In the present paper, we prove nonexistence results for the following nonlinear evolution equation, see works of T. Cazenave and A. Haraux (1990) and S. Zheng (2004), u t t + f ( x ) u t + ( - Δ ) α / 2 ( u m ) = h ( t , x ) | u | p , posed in ( 0 , T ) × ℝ N , where ( - Δ ) α / 2 , 0 < α ≤ 2 is α / 2 -fractional power of - Δ . Our method of proof is based on suitable choices of the test functions in the weak formulation of the sought solutions. Then, we extend this result to the case of a 2 × 2 system of the same type.

How to cite

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Djilali, Medjahed, and Hakem, Ali. "Results of nonexistence of solutions for some nonlinear evolution problems." Commentationes Mathematicae Universitatis Carolinae 60.2 (2019): 269-284. <http://eudml.org/doc/294828>.

@article{Djilali2019,
abstract = {In the present paper, we prove nonexistence results for the following nonlinear evolution equation, see works of T. Cazenave and A. Haraux (1990) and S. Zheng (2004), \[ u\_\{tt\} +f(x)u\_t +(-\Delta )^\{\alpha /2\}(u^m)= h(t,x) |u|^\{p\}, \] posed in $(0,T)\times \mathbb \{R\}^\{N\},$ where $(-\Delta )^\{\{\alpha \}/\{2\}\}, 0<\alpha \le 2$ is $\{\alpha \}/\{2\}$-fractional power of $\,-\Delta .$ Our method of proof is based on suitable choices of the test functions in the weak formulation of the sought solutions. Then, we extend this result to the case of a $2\times 2$ system of the same type.},
author = {Djilali, Medjahed, Hakem, Ali},
journal = {Commentationes Mathematicae Universitatis Carolinae},
keywords = {nonexistence; test functions; global weak solution; fractional Laplacian; critical exponent},
language = {eng},
number = {2},
pages = {269-284},
publisher = {Charles University in Prague, Faculty of Mathematics and Physics},
title = {Results of nonexistence of solutions for some nonlinear evolution problems},
url = {http://eudml.org/doc/294828},
volume = {60},
year = {2019},
}

TY - JOUR
AU - Djilali, Medjahed
AU - Hakem, Ali
TI - Results of nonexistence of solutions for some nonlinear evolution problems
JO - Commentationes Mathematicae Universitatis Carolinae
PY - 2019
PB - Charles University in Prague, Faculty of Mathematics and Physics
VL - 60
IS - 2
SP - 269
EP - 284
AB - In the present paper, we prove nonexistence results for the following nonlinear evolution equation, see works of T. Cazenave and A. Haraux (1990) and S. Zheng (2004), \[ u_{tt} +f(x)u_t +(-\Delta )^{\alpha /2}(u^m)= h(t,x) |u|^{p}, \] posed in $(0,T)\times \mathbb {R}^{N},$ where $(-\Delta )^{{\alpha }/{2}}, 0<\alpha \le 2$ is ${\alpha }/{2}$-fractional power of $\,-\Delta .$ Our method of proof is based on suitable choices of the test functions in the weak formulation of the sought solutions. Then, we extend this result to the case of a $2\times 2$ system of the same type.
LA - eng
KW - nonexistence; test functions; global weak solution; fractional Laplacian; critical exponent
UR - http://eudml.org/doc/294828
ER -

References

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