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On a parabolic boundary value problem. (English. Russian original) Zbl 0766.35019

Sov. Math., Dokl. 43, No. 2, 334-338 (1991); translation from Dokl. Akad. Nauk SSSR 317, No. 1, 39-43 (1991).
From the authors’ introduction: “In the rectangle \(Q_ T=\{(t,x)\): \(0<t<T\), \(0<x<1\}\) we consider the following problem: \[ u_ t=k(t)u_{xx},\quad (t,x)\in Q_ T; \qquad u(0,x)=u_ 0(x),\quad x\in(0,1); \qquad u_ x(t,0)=0,\quad t\in(0,T); \]
\[ p_ 0 u_ 0(1)- \int_ 0^ 1 k(t')u_ x(t',1)dt'=p(t)u(t,1), \qquad t\in(0,T), \] where \(u_ 0(x)\in C[0,1]\), \(k(t)\in L_ \infty(0,T)\), and \(p(t)\in L_ \infty(0,T)\) are given functions with (a.e. on \((0,T)\)), \(0<k'\leq k(t)\leq k''<\infty\) and \(0<p'\leq p(t)\leq p''<\infty\); \(k'\), \(k''\), \(p''\), and \(T\) are given positive numbers.”
The paper gives some results on the existence and uniqueness of the solution to the boundary value problem mentioned above.
Reviewer: M.Chicco (Genova)

MSC:

35K20 Initial-boundary value problems for second-order parabolic equations
35Q80 Applications of PDE in areas other than physics (MSC2000)
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