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Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term

  • Author Bio: E-mail address: svernier@unica.it; E-mail address: giuseppe.viglialoro@unica.it
  • Monica Marras, E-mail address: mmarras@unica.it

    Monica Marras, E-mail address: mmarras@unica.it 
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  • This paper is concerned with the pseudo-parabolic problem

    $\left\{ \begin{array}{l}\begin{split}u_t- \lambda \triangle u_t=& k(t) \text{div}(g(| \nabla u|^2) \nabla u) +f(t,u,| \nabla u| ) \quad {\rm in} \ \Omega \times (0, t^*), \\[6pt] u=&0 \ \qquad {\rm on} \ \partial \Omega \times (0,t^*),\\[6pt] u ({ x},0) =& u_0 ({ x}) \quad {\rm in} \ \Omega,\\[6pt]\end{split}\end{array} \right.$

    where $\Omega$ is a bounded domain in $\mathbb{R}^n, \ n\geq 2$, with smooth boundary $ \partial \Omega$, $ k$ is a positive constant or in general positive derivable function of $t$. The solution $u(x,t)$ may or may not blow up in finite time. Under suitable conditions on data, a lower bound for $t^*$ is derived, where $[0,t^*)$ is the time interval of existence of $u(x,t).$ We indicate how some of our results can be extended to a class of nonlinear pseudo-parabolic systems.

    Mathematics Subject Classification: Primary:35K70, 35B44;Secondary:35B44.


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  • [1] A. B. Al'Shin, M. O. Korpusov and A. G. Sveshnikov, Blow-up in Nonlinear Sobolev Type Equations, De Gruyter Series in Nonlinear Analysis and Applications, 2011.

    G. I. Barenblatt, I. P. Zeltov and I. N. Kockina, Basic concepts in the theory of seepage, J. Sov. Appl. Math. Mech., 24 (1960), 852-864.


    G. I. Barenblatt, Yu. P. Zheltov and I. N. Kochina, Foundations of filtration theory in cracked media, Appl. Math. Mech., 24 (1960), 58-73.


    P. J. Chen and M. E. Gurtin, On a theory of heat conduction involving two temperatures, Z. Anghew. Math. Phys., 19 (1968), 614-627.


    E. Di Benedetto and M. Pierre, On the maximum principle for pseudoparabolic Equations, Indiana Univ. Math. J., 30 (1981), 821-854.


    H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form Put = -Au + F(u), Arch. Rational Mech. Anal., 51 (1973), 371-386.


    P. Luo, Blow-up phenomena for a pseudo-parabolic equation, Math. Meth. Appl. Sci., 38 (2015), 2636-2641.


    M. Marras and S. Vernier Piro, On global existence and bounds for blow-up time in non linear parabolic problems with time dependent coefficients, Discrete Contin. Dyn. Syst., 2013 (2013), 535-544.


    M. Marras and S. Vernier-Piro, Blow up and decay bounds in quasilinear parabolic problems, Discrete Contin. Dyn. Syst., 2007 (2007), 704-712.


    M. Marras, S. Vernier-Piro and G. Viglialoro, Estimates from below of blow-up time in a parabolic system with gradient term, International Journal of Pure and Applied Mathematics, 93 (2014), 297-306.


    M. Marras, S. Vernier-Piro and G. Viglialoro, Lower bounds for blow-up time in a parabolic problem with a gradient term under various boundary conditions, Kodai Mathematical Journal, 37 (2014), 532-543.


    G. A. Philippin, Lower bounds for blow-up time in a class of nonlinear wave equations, Z. Angew. Math. Phys., 66 (2015), 129-134.


    R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26.


    S. L. Sobolev, On a new problem of mathematical physics, Izv. Akad. Nauk SSSR Ser. Mat., 18 (1954), 3-50.


    T. W. Ting, Parabolic and pseudo-parabolic partial differential equations, J. Math. Soc. Japan, 21 (1969) 440-453.


    G. Viglialoro, On the blow-up time of a parabolic system with damping terms, Comptes Rendus de L'Academie Bulgare des Sciences, 67 (2014), 1223-1232.


    R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudoparabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.

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