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Mat. Zametki, 1996, Volume 60, Issue 3, Pages 356–362 (Mi mz1835)  

This article is cited in 21 scientific papers (total in 21 papers)

Unique solvability of the Cauchy problem for certain quasilinear pseudoparabolic equations

A. L. Gladkov

Vitebsk State University named after P. M. Masherov

Abstract: 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'(u(x,t)) &\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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English version:
Mathematical Notes, 1996, 60:3, 264–268

Bibliographic databases:

UDC: 517.956
Received: 09.07.1993

Citation: A. L. Gladkov, “Unique solvability of the Cauchy problem for certain quasilinear pseudoparabolic equations”, Mat. Zametki, 60:3 (1996), 356–362; Math. Notes, 60:3 (1996), 264–268

Citation in format AMSBIB
\by A.~L.~Gladkov
\paper Unique solvability of the Cauchy problem for certain quasilinear pseudoparabolic equations
\jour Mat. Zametki
\yr 1996
\vol 60
\issue 3
\pages 356--362
\jour Math. Notes
\yr 1996
\vol 60
\issue 3
\pages 264--268

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    This publication is cited in the following articles:
    1. M. O. Korpusov, “Blow-up of a solution of a pseudoparabolic equation with the time derivative of a nonlinear elliptic operator”, Comput. Math. Math. Phys., 42:12 (2002), 1717–1724  mathnet  mathscinet  zmath
    2. M. O. Korpusov, “Global solvability of pseudoparabolic nonlinear equations and blow-up of their solutions”, Comput. Math. Math. Phys., 42:6 (2002), 814–831  mathnet  mathscinet  zmath
    3. M. O. Korpusov, A. G. Sveshnikov, “Energy estimate of the solution to a nonlinear pseudoparabolic equation at $t\to\infty$”, Comput. Math. Math. Phys., 42:8 (2002), 1154–1160  mathnet  mathscinet  zmath  elib
    4. M. O. Korpusov, A. G. Sveshnikov, “Three-dimensional nonlinear evolution equations of pseudoparabolic type in problems of mathematical physics”, Comput. Math. Math. Phys., 43:12 (2003), 1765–1797  mathnet  mathscinet  zmath
    5. M. O. Korpusov, “Conditions for the global solvability of the Cauchy problem for a semilinear equation of pseudoparabolic type”, Comput. Math. Math. Phys., 43:8 (2003), 1159–1171  mathnet  mathscinet  zmath
    6. M. O. Korpusov, A. G. Sveshnikov, “On the solvability of strongly nonlinear pseudoparabolic equation with double nonlinearity”, Comput. Math. Math. Phys., 43:7 (2003), 944–961  mathnet  mathscinet  zmath  elib
    7. M. O. Korpusov, A. G. Sveshnikov, “On the existence of a solution to the Laplace equation with a nonlinear dynamic boundary condition”, Comput. Math. Math. Phys., 43:1 (2003), 92–107  mathnet  mathscinet  zmath
    8. M. O. Korpusov, A. G. Sveshnikov, “On the blow-up of the solution of an initial-boundary value problem for a nonlinear nonlocal equation of pseudo-parabolic type”, Comput. Math. Math. Phys., 44:12 (2004), 2104–2111  mathnet  mathscinet  zmath
    9. Korpusov, MO, “Global solvability conditions for an initial-boundary value problem for a nonlinear equation of pseudoparabolic type”, Differential Equations, 41:5 (2005), 712  mathnet  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    10. M. O. Korpusov, A. G. Sveshnikov, “On the finite-time blowup of solutions to initial–boundary value problems for pseudoparabolic equations with pseudo-Laplacian”, Comput. Math. Math. Phys., 45:2 (2005), 261–274  mathnet  mathscinet  zmath  elib  elib
    11. M. O. Korpusov, A. G. Sveshnikov, ““Destruction” of the solution of a strongly nonlinear equation of pseudoparabolic type with double nonlinearity”, Math. Notes, 79:6 (2006), 820–840  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    12. E. I. Kaikina, P. I. Naumkin, I. A. Shishmarev, “Large-time asymptotic behaviour of solutions of non-linear Sobolev-type equations”, Russian Math. Surveys, 64:3 (2009), 399–468  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
    13. E. I. Kaikina, P. I. Naumkin, I. A. Shishmarev, “Periodic Boundary Value Problem for Nonlinear Sobolev-Type Equations”, Funct. Anal. Appl., 44:3 (2010), 171–181  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    14. Li Y., Cao Ya., Yin J., Wang Y., “Time Periodic Solutions for a Viscous Diffusion Equation with Nonlinear Periodic Sources”, Electron. J. Qual. Theory Differ., 2011, no. 10, 1–19  mathscinet  isi
    15. Cao Ya., Yin J., Jin Ch., “A Periodic Problem of a Semilinear Pseudoparabolic Equation”, Abstract Appl. Anal., 2011, 363579  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    16. Kavitova T.V., “Suschestvovanie resheniya zadachi koshi dlya psevdoparabolicheskogo uravneniya”, Vesnk Vtsebskaga dzyarzhainaga unversteta, 3:63 (2011), 14–19  elib
    17. Kavitova T., “Behavior of the Maximal Solution of the Cauchy Problem for Some Nonlinear Pseudoparabolic Equation as Vertical Bar X Vertical Bar -> Infinity”, Electron. J. Differ. Equ., 2012, 141  mathscinet  zmath  isi  elib
    18. Li Y., Cao Ya., Yin J., “A Class of Viscous P-Laplace Equation with Nonlinear Sources”, Chaos Solitons Fractals, 57 (2013), 24–34  crossref  mathscinet  zmath  adsnasa  isi  elib  scopus  scopus
    19. Li Y., Cao Ya., “Time Periodic Solutions For a Pseudo-Parabolic Type Equation With Weakly Nonlinear Periodic Sources”, Bull. Malays. Math. Sci. Soc., 38:2 (2015), 667–682  crossref  mathscinet  zmath  isi  scopus  scopus
    20. S. G. Pyatkov, S. N. Shergin, “On some mathematical models of filtration theory”, Vestn. YuUrGU. Ser. Matem. modelirovanie i programmirovanie, 8:2 (2015), 105–116  mathnet  crossref  elib
    21. A. A. Kon'kov, A. E. Shishkov, “On the Absence of Global Solutions of a Class of Higher-Order Evolution Inequalities”, Math. Notes, 104:6 (2018), 930–932  mathnet  crossref  crossref  isi  elib
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