`a`
Communications on Pure and Applied Analysis (CPAA)
 

Global existence and blow up of solutions to a class of pseudo-parabolic equations with an exponential source
Pages: 2465 - 2485, Issue 6, November 2015

doi:10.3934/cpaa.2015.14.2465      Abstract        References        Full text (475.7K)           Related Articles

Xiaoli Zhu - School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, China (email)
Fuyi Li - School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, China (email)
Ting Rong - School of Mathematical Sciences, Shanxi University, Taiyuan 030006, Shanxi, China (email)

1 C. O. Alves and M. M. Cavalcanti, On existence, uniform decay rates and blow up for solutions of the 2-D wave equation with exponential source, Calc. Var. Partial Differential Equations, 34 (2009), 377-411.       
2 H. Brill, A semilinear Sobolev evolution equation in a Banach space, J. Differential Equations, 24 (1977), 412-425.       
3 Y. Cao, J. Yin and C. Wang, Cauchy problems of semilinear pseudo-parabolic equations, J. Differential Equations, 246 (2009), 4568-4590.       
4 T. Cazenave and A. Haraux, An Introduction to Semilinear Evolution Equations, vol. 13 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press, Oxford University Press, New York, 1998,       
5 H. Chen and S. Tian, Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity, J. Differential Equations, 258 (2015), 4424-4442.       
6 D. Colton, Pseudoparabolic equations in one space variable, J. Differential Equations, 12 (1972), 559-565.       
7 D. Colton and J. Wimp, Asymptotic behaviour of the fundamental solution to the equation of heat conduction in two temperatures, J. Math. Anal. Appl., 69 (1979), 411-418.       
8 D. G. de Figueiredo, O. H. Miyagaki and B. Ruf, Elliptic equations in $R^2$ with nonlinearities in the critical growth range, Calc. Var. Partial Differential Equations, 3 (1995), 139-153.       
9 E. DiBenedetto and M. Pierre, On the maximum principle for pseudoparabolic equations, Indiana Univ. Math. J., 30 (1981), 821-854.       
10 F. Gazzola and M. Squassina, Global solutions and finite time blow up for damped semilinear wave equations, Ann. Inst. H. Poincaré Anal. Non Linéaire, 23 (2006), 185-207.       
11 M. O. Korpusov and A. G. Sveshnikov, Blow-up of solutions of strongly nonlinear equations of pseudoparabolic type, J. Math. Sci. (N. Y.), 148 (2008), 1-142.       
12 H. A. Levine, Some nonexistence and instability theorems for solutions of formally parabolic equations of the form $Pu_t=-Au+F(u)$, Arch. Rational Mech. Anal., 51 (1973), 371-386.       
13 Y. Liu, R. Xu and T. Yu, Global existence, nonexistence and asymptotic behavior of solutions for the Cauchy problem of semilinear heat equations, Nonlinear Anal., 68 (2008), 3332-3348.       
14 M. Meyvaci, Blow up of solutions of pseudoparabolic equations, J. Math. Anal. Appl., 352 (2009), 629-633.       
15 J. Moser, A sharp form of an inequality by N. Trudinger, Indiana Univ. Math. J., 20 (1970/71), 1077-1092.       
16 V. Padrón, Effect of aggregation on population revovery modeled by a forward-backward pseudoparabolic equation, Trans. Amer. Math. Soc., 356 (2004), 2739-2756 (electronic).       
17 L. E. Payne and D. H. Sattinger, Saddle points and instability of nonlinear hyperbolic equations, Israel J. Math., 22 (1975), 273-303.       
18 M. Peszyńska, R. Showalter and S.-Y. Yi, Homogenization of a pseudoparabolic system, Appl. Anal., 88 (2009), 1265-1282.       
19 W. Rundell, The construction of solutions to pseudoparabolic equations in noncylindrical domains, J. Differential Equations, 27 (1978), 394-404.       
20 D. H. Sattinger, On global solution of nonlinear hyperbolic equations, Arch. Rational Mech. Anal., 30 (1968), 148-172.       
21 N. Seam and G. Vallet, Existence results for nonlinear pseudoparabolic problems, Nonlinear Anal. Real World Appl., 12 (2011), 2625-2639.       
22 R. E. Showalter, Existence and representation theorems for a semilinear Sobolev equation in Banach space, SIAM J. Math. Anal., 3 (1972), 527-543.       
23 R. E. Showalter, Weak solutions of nonlinear evolution equations of Sobolev-Galpern type, J. Differential Equations, 11 (1972), 252-265.       
24 R. E. Showalter and T. W. Ting, Pseudoparabolic partial differential equations, SIAM J. Math. Anal., 1 (1970), 1-26.       
25 T. W. Ting, Certain non-steady flows of second-order fluids, Arch. Rational Mech. Anal., 14 (1963), 1-26.       
26 N. S. Trudinger, On imbeddings into Orlicz spaces and some applications, J. Math. Mech., 17 (1967), 473-483.       
27 M. Willem, Minimax Theorems, Progress in Nonlinear Differential Equations and Their Applications, 24, Birkhäuser Boston Inc., Boston, MA, 1996.       
28 R. Xu and J. Su, Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations, J. Funct. Anal., 264 (2013), 2732-2763.       
29 C. Yang, Y. Cao and S. Zheng, Second critical exponent and life span for pseudo-parabolic equation, J. Differential Equations, 253 (2012), 3286-3303.       
30 X. Zhu, F. Li and Y. Li, Some sharp result about the global existence and blow up of solutions to a class of pseudo-parabolic equations, preprint.

Go to top