Geodesic
Unique solvability of the Cauchy problem for certain quasilinear pseudoparabolic equations
Matematičeskie zametki, Tome 60 (1996) no. 3, pp. 356-362.

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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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     author = {A. L. Gladkov},
     title = {Unique solvability of the {Cauchy} problem for certain quasilinear pseudoparabolic equations},
     journal = {Matemati\v{c}eskie zametki},
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     publisher = {mathdoc},
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     number = {3},
     year = {1996},
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     url = {http://geodesic.mathdoc.fr/item/MZM_1996_60_3_a2/}
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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/

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