A higher order evolution inequality with a gradient term in the exterior of the half-ball

Ibtehal Alazman; Ibtisam Aldawish; Mohamed Jleli; Bessem Samet; Ibtehal Alazman; Ibtisam Aldawish; Mohamed Jleli; Bessem Samet

doi:10.3934/math.2023463

AIMS Mathematics

2023, Volume 8, Issue 4: 9230-9246. doi: 10.3934/math.2023463

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A higher order evolution inequality with a gradient term in the exterior of the half-ball

1.
Department of Mathematics and Statistics, College of Science, Imam Mohammad Ibn Saud Islamic University (IMSIU), Riyadh 11566, Saudi Arabia
2.
Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

Received: 29 October 2022 Revised: 18 January 2023 Accepted: 29 January 2023 Published: 14 February 2023
MSC : 35R45, 35B44, 35B33

We study the existence and nonexistence of weak solutions to a semilinear higher order (in time) evolution inequality involving a convection term in the exterior of the half-ball, under Dirichlet-type boundary conditions. A weight function of the form $ |x|^a $ is allowed in front of the power nonlinearity. When $ a > -2 $, we show that the dividing line with respect to existence or nonexistence is given by a critical exponent (Fujita critical exponent), which depends on the parameters of the problem, but independent of the order of the time-derivative. Our study yields naturally optimal nonexistence results for the corresponding stationary problem.
- evolution inequality,
- convection term,
- half-ball,
- critical exponent
Citation: Ibtehal Alazman, Ibtisam Aldawish, Mohamed Jleli, Bessem Samet. A higher order evolution inequality with a gradient term in the exterior of the half-ball[J]. AIMS Mathematics, 2023, 8(4): 9230-9246. doi: 10.3934/math.2023463

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Abstract

We study the existence and nonexistence of weak solutions to a semilinear higher order (in time) evolution inequality involving a convection term in the exterior of the half-ball, under Dirichlet-type boundary conditions. A weight function of the form $ |x|^a $ is allowed in front of the power nonlinearity. When $ a > -2 $, we show that the dividing line with respect to existence or nonexistence is given by a critical exponent (Fujita critical exponent), which depends on the parameters of the problem, but independent of the order of the time-derivative. Our study yields naturally optimal nonexistence results for the corresponding stationary problem.

References

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