Electron. J. Differential Equations, Vol. 2018 (2018), No. 09, pp. 1-13.

Nonexistence of global solutions to the system of semilinear parabolic equations with biharmonic operator and singular potential

Shirmayil Bagirov

Abstract:
In the domain $Q_{R}'= \{ x:| x| >R\}\times( 0,+\infty)$ we consider the problem
$$\displaylines{ \frac{\partial u_1}{\partial t}+\Delta^2 u_1-\frac{C_1}{|x| ^4}u_1 =| x| ^{\sigma _1}| u_2| ^{q_1}, \quad u_1| _{t=0}=u_{10}( x)\geq0, \cr \frac{\partial u_2}{\partial t}+\Delta^2 u_2-\frac{C_2}{| x| ^4}u_2=| x| ^{\sigma _2}| u_1| ^{q_2},\quad u_2| _{t=0}=u_{20}( x)\geq0, \cr \int_0^\infty \int_{\partial B_{R}} u_i\,ds\,dt\geq 0, \quad \int_0^\infty \int_{\partial B_{R}}\Delta u_i\,ds\,dt\leq 0, }$$
where $\sigma_i\in \mathbb{R} $, $ q_i>1 $, $ 0\leq C_i<( \frac{n( n-4) }{4}) ^2$, $ i=1,2 $. Sufficient condition for the nonexistence of global solutions is obtained.The proof is based on the method of test functions.

Submitted November 12, 2017. Published January 6, 2018.
Math Subject Classifications: 35A01, 35B33, 35K52, 35K91.
Key Words: System of semilinear parabolic equation; biharmonic operator; global solution; critical exponent; method of test functions.

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Shirmayil Bagirov
Institute of Mathematics and Mechanics of NAS of Azerbaijan
Baku, Azerbaijan
email: sh_bagirov@yahoo.com

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