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

• Main Author: Shirmayil Bagirov
• Format: Article
• Language: English
• Published: Texas State University 2018-01-01
• Series: Electronic Journal of Differential Equations
• Subjects: System of semilinear parabolic equation; biharmonic operator; global solution; critical exponent; method of test functions
• Online Access: http://ejde.math.txstate.edu/Volumes/2018/09/abstr.html

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.