# zbMATH — the first resource for mathematics

The process of heat conductivity with autoregulating impulse support. (Ukrainian) Zbl 1004.35119
The article deals with the parabolic differential equation $$\frac{\partial u}{\partial t}=a\Delta u(x,t), (x,t) \in \Omega\times {\mathbb R}_+,$$ $$u(x,t)=0, (x,t)\in \partial\Omega\times {\mathbb R}_+, u(x,0)=u_0(x), x\in\Omega,$$ under the condition of impulse influence $$(u(x,t^+)-u(x,t^-)) |_{I_u(t)=I_0}= \alpha(x),$$ where $$\alpha(x)$$ is a known function, $$I_u(t) = \int_{\Omega}u(x,t) dx$$ is a regularizing functional, $$u_0\in C(\Omega, {\mathbb R}_+),$$ $$\Omega=[0,l_1]\times\dots \times [0,l_n].$$ The author proves existence of an infinite sequence of the impulsive moments. Conditions of existence and uniqueness of a periodic solution to the original problem are proposed.
##### MSC:
 35R12 Impulsive partial differential equations 35K20 Initial-boundary value problems for second-order parabolic equations