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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
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