Geodesic
A Nonlinear Loaded Parabolic Equation and a Related Inverse Problem
Matematičeskie zametki, Tome 76 (2004) no. 6, pp. 840-853

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The solvability of the nonlocal-in-time boundary-value problem for the nonlinear parabolic equation $u_t-\Delta u+c(\bar u(x,T))u=f(x,t)$, where $\bar u(x,t)= \alpha(t)u(x,t)+\int^t_0\beta(\tau)u(x,\tau)\,d\tau$ for given functions $\alpha(t)$ and $\beta(t)$, is studied. Existence and uniqueness theorems for regular solutions are proved; it is shown that the results obtained can be used to study the solvability of coefficient inverse problems. 
@article{MZM_2004_76_6_a4,
     author = {A. I. Kozhanov},
     title = {A {Nonlinear} {Loaded} {Parabolic} {Equation} and a {Related} {Inverse} {Problem}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {840--853},
     publisher = {mathdoc},
     volume = {76},
     number = {6},
     year = {2004},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2004_76_6_a4/}
}
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A. I. Kozhanov. A Nonlinear Loaded Parabolic Equation and a Related Inverse Problem. Matematičeskie zametki, Tome 76 (2004) no. 6, pp. 840-853. http://geodesic.mathdoc.fr/item/MZM_2004_76_6_a4/