Classification of initial energy in a pseudo-parabolic equation with variable exponents

This paper deals with a pseudo-parabolic equation with variable exponents, subject to homogeneous boundary conditions with initial data in $$H_0^1(\varOmega )$$ . By using energy functional and Nehari functional, we classify blow-up and global existence of weak solutions in variable Sobolev spaces completely for subcritical, critical, and super critical initial energy, respectively.

[1]  Juan Pablo Pinasco,et al.  Blow-up for parabolic and hyperbolic problems with variable exponents , 2009 .

[2]  Xiaoming Peng,et al.  Blow-up phenomena for a pseudo-parabolic equation with variable exponents , 2017, Appl. Math. Lett..

[3]  Heinz Brill,et al.  A semilinear Sobolev evolution equation in a Banach space , 1977 .

[4]  Bingchen Liu,et al.  Asymptotic analysis for blow-up solutions in parabolic equations involving variable exponents , 2013 .

[5]  Yuzhu Han,et al.  Finite time blowup for a semilinear pseudo-parabolic equation with general nonlinearity , 2020, Appl. Math. Lett..

[6]  Yunmei Chen,et al.  Variable Exponent, Linear Growth Functionals in Image Restoration , 2006, SIAM J. Appl. Math..

[7]  Giuseppe Mingione,et al.  Regularity Results for Stationary Electro-Rheological Fluids , 2002 .

[8]  G. I. Barenblatt,et al.  Basic concepts in the theory of seepage of homogeneous liquids in fissured rocks [strata] , 1960 .

[9]  Bin Guo,et al.  Global existence and blow-up of weak solutions for a pseudo-parabolic equation with high initial energy , 2020, Appl. Math. Lett..

[10]  Shuying Tian,et al.  Initial boundary value problem for a class of semilinear pseudo-parabolic equations with logarithmic nonlinearity , 2015 .

[11]  Jingxue Yin,et al.  Cauchy problems of semilinear pseudo-parabolic equations ✩ , 2009 .

[12]  Runzhang Xu,et al.  Global existence and finite time blow-up for a class of semilinear pseudo-parabolic equations , 2013 .

[13]  J. Rodrigues,et al.  On stationary thermo-rheological viscous flows , 2006 .

[14]  M. Ružička,et al.  Mathematical modeling of electrorheological materials , 2001 .

[15]  Bingchen Liu,et al.  A complete classification of initial energy in a p(x)-Laplace pseudo-parabolic equation , 2021, Appl. Math. Lett..

[16]  Yonghong Wu,et al.  Finite time blow-up for a class of parabolic or pseudo-parabolic equations , 2018, Comput. Math. Appl..

[17]  W. Lian,et al.  Global existence and blow up of solutions for pseudo-parabolic equation with singular potential , 2020, Journal of Differential Equations.

[18]  J. D. Rossi,et al.  Critical exponents for a semilinear parabolic equation with variable reaction , 2012, Proceedings of the Royal Society of Edinburgh: Section A Mathematics.

[19]  Menglan Liao.  Non-global existence of solutions to pseudo-parabolic equations with variable exponents and positive initial energy , 2019, Comptes Rendus Mécanique.