Differential equations and Optimal control

On blow-up set of solutions of initial boundary value problem for a system of parabolic equations with nonlocal boundary conditions

A. Gladkov^a, A. I. Nikitin<sup>b

a</sup> Belarusian State University, 4 Niezaliežnasci Avenue, Minsk 220030, Belarus
^b Vitebsk State University named after P. M. Masherov, 33 Maskouski Avenue, Vitebsk 210038, Belarus

Abstract: We consider a system of semilinear parabolic equations $u_{t}=\Delta u + c_{1}(x,t)\nu^{p}, \nu_{t}=\Delta\nu+c_{2}(x,t)u^{q}, (x,t)\in \Omega\times (0,+\infty)$ with nonlinear nonlocal boundary conditions $\dfrac{\partial u}{\partial\eta}= \int\limits_\Omega k_{1}(x,y,t)u^{m}(y,t)dy, \dfrac{\partial\nu}{\partial\eta}= \int\limits_\Omega k_{2}(x,y,t)\nu^{n}(y,t)dy, (x,t)\in \partial\Omega\times (0,+\infty)$ and initial data $u(x,0)=u_{0}(x), \nu(x,0)=\nu_{0}(x), x\in \Omega$, where $p,q,m,n$ are positive constants, $\Omega$ is bounded domain in $\mathbb{R}^{N}(N\geq 1)$ with a smooth boundary $\partial\Omega, \eta$ is unit outward normal on $\partial\Omega$. Nonnegative locally Holder continuous functions $c_{i}(x,t), i=1,2$, are defined for $x\in \overline{\Omega}, t\geq 0$; nonnegative continuous functions $k_{i}(x,y,t), i=1,2$ are defined for $x\in \partial\Omega, y\in \overline{\Omega}, t\geq 0$; nonnegative continuous functions $u_{0}(x), \nu_{0}(x)$ are defined for $x\in \overline{\Omega}$ and satisfy the conditions $\dfrac{\partial u_{0}(x)}{\partial\eta}= \int\limits_\Omega k_{1}(x,y,0)u_{0}^{m}(y)dy, \dfrac{\partial \nu_{0}(x)}{\partial\eta}= \int\limits_\Omega k_{2}(x,y,0)\nu_{0}^{n}(y)dy$ for $x\in \partial\Omega$. In the paper blow-up set of classical solutions is investigated. It is established that blow-up of the solutions can occur only on the boundary $\partial\Omega$ if $max(p,q)\leq 1, max(m,n)>1$ and under certain conditions for the coefficients $k_{i}(x,y,t), i=1,2$.

Keywords: system of semilinear parabolic equations, nonlocal boundary conditions, blow-up set.

UDC: 517.95

Received: 20.03.2018