Matematičeskie zametki, Tome 60 (1996) no. 3, pp. 356-362
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A. L. Gladkov. Unique solvability of the Cauchy problem for certain quasilinear pseudoparabolic equations. Matematičeskie zametki, Tome 60 (1996) no. 3, pp. 356-362. http://geodesic.mathdoc.fr/item/MZM_1996_60_3_a2/
@article{MZM_1996_60_3_a2,
author = {A. L. Gladkov},
title = {Unique solvability of the {Cauchy} problem for certain quasilinear pseudoparabolic equations},
journal = {Matemati\v{c}eskie zametki},
pages = {356--362},
year = {1996},
volume = {60},
number = {3},
language = {ru},
url = {http://geodesic.mathdoc.fr/item/MZM_1996_60_3_a2/}
}
TY - JOUR
AU - A. L. Gladkov
TI - Unique solvability of the Cauchy problem for certain quasilinear pseudoparabolic equations
JO - Matematičeskie zametki
PY - 1996
SP - 356
EP - 362
VL - 60
IS - 3
UR - http://geodesic.mathdoc.fr/item/MZM_1996_60_3_a2/
LA - ru
ID - MZM_1996_60_3_a2
ER -
%0 Journal Article
%A A. L. Gladkov
%T Unique solvability of the Cauchy problem for certain quasilinear pseudoparabolic equations
%J Matematičeskie zametki
%D 1996
%P 356-362
%V 60
%N 3
%U http://geodesic.mathdoc.fr/item/MZM_1996_60_3_a2/
%G ru
%F MZM_1996_60_3_a2
We study the Cauchy problem in the layer $\Pi_T={\mathbb R}^n\times[0,T]$ for the equation $$ u_t=c\Delta u_t+\Delta\varphi(u), $$ where $c$ is a positive constant and the function $\varphi(p)$ belongs to $C^1({\mathbb R}_+)$ and has a nonnegative monotone non-decreasing derivative. The unique solvability of this Cauchy problem is established for the class of nonnegative functions $u(x,t)\in C_{x,t}^{2,1}(\Pi_T)$ with the properties: \begin{align*} \varphi'\bigl(u(x,t)\bigr) &\le M_1(1+|x|^2), \\ \|u_t(x,t)| & \le M_2(1+|x|^2)^\beta\qquad (\beta >0). \end{align*}
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