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August  2017, 22(6): 2291-2300. doi: 10.3934/dcdsb.2017096

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

Monica Marras , , Stella Vernier-Piro and  Giuseppe Viglialoro

Università di Cagliari, Dipartimento di Matematica e Informatica, V.le Merello 92,09123 Cagliari, Italy

Monica Marras, E-mail address: mmarras@unica.it

Received  April 2016 Revised  December 2016 Published  March 2017

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.
Keywords: Pseudo-parabolic equations, pseudo-parabolic systems, blow-up, global existence.
Mathematics Subject Classification: Primary: 35K70, 35B44; Secondary: 35B4.
Citation: Monica Marras, Stella Vernier-Piro, Giuseppe Viglialoro. Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term. Discrete & Continuous Dynamical Systems - B, 2017, 22 (6) : 2291-2300. doi: 10.3934/dcdsb.2017096
References:
[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. Google Scholar

[2]

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.   Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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.   Google Scholar

[7]

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

[8]

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.   Google Scholar

[9]

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

[10]

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.   Google Scholar

[11]

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.   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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.   Google Scholar

[17]

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.   Google Scholar

show all references

References:
[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. Google Scholar

[2]

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.   Google Scholar

[3]

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

[4]

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

[5]

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

[6]

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.   Google Scholar

[7]

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

[8]

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.   Google Scholar

[9]

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

[10]

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.   Google Scholar

[11]

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.   Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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.   Google Scholar

[17]

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.   Google Scholar

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