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Blow-up phenomena for nonlinear pseudo-parabolic equations with gradient term
Università di Cagliari, Dipartimento di Matematica e Informatica, V.le Merello 92,09123 Cagliari, Italy |
$\left\{ \begin{array}{l}\begin{split}u_t- \lambda \triangle u_t=& k(t) \text{div}(g(| \nabla u|^2) \nabla u) +f(t,u,| \nabla u| ) \quad {\rm in} \ \Omega \times (0, t^*), \\[6pt] u=&0 \ \qquad {\rm on} \ \partial \Omega \times (0,t^*),\\[6pt] u ({ x},0) =& u_0 ({ x}) \quad {\rm in} \ \Omega,\\[6pt]\end{split}\end{array} \right.$ |
$\Omega$ |
$\mathbb{R}^n, \ n\geq 2$ |
$ \partial \Omega$ |
$ k$ |
$t$ |
$u(x,t)$ |
$t^*$ |
$[0,t^*)$ |
$u(x,t).$ |
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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