Qualitative Study in a Parabolic Equation with Nonstandard Growth Conditions and Singular Medium Void

Acerbi, E., Mingione, G.: Regularity results for stationary electro-rheological fluids. Arch. Ration. Mech. Anal. 164, 213–259 (2002)

Antontsev, S., Rodrigues, J.F.: On stationary thermo-rheological viscous flows. Ann. Univ. Ferrara Sez. VII Sci. Mat. 52, 19–36 (2006)

Aboulaich, R., Meskine, D., Souissi, A.: New diffusion models in image processing. Comput. Math. Appl. 56, 874–882 (2008)

Antontsev, S., Shmarev, S.: Parabolic equations with anisotropic nonstandard growth conditions. Int. Ser. Numer. Math. 154, 33–44 (2007)

Antontsev, S., Shmarev, S.: Anisotropic parabolic equations with variable nonlinearity. Publ. Mat. 53, 355–399 (2009)

Antontsev, S., Shmarev, S.: Extinction of solutions of parabolic equations with variable anisotropic nonlinearities. Proc. Steklov Inst. Math. 261, 11–22 (2008)

Antontsev, S., Shmarev, S.: Vanishing solutions of anisotropic parabolic equations with variable nonlinearity. J. Math. Anal. Appl. 361, 371–391 (2010)

Badiale, M., Tarantello, G.A.: Sobolev–Hardy inequality with applications to a nonlinear elliptic equation arising in astrophysics. Arch. Ration. Mech. Anal. 163, 259–293 (2002)

Boccardo, L., Murat, F.: Almost everywhere convergence of the gradients of solutions to elliptic and parabolic equations. Nonlinear Anal. 19, 581–597 (1992)

Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, New York (2011)

Chen, Y.M., Levine, S.E., Rao, M.: Variable exponent, linear growth functionals in image restoration. SIAM J. Appl. Math. 66, 1383–1406 (2006)

Day, W.A.: Existence of a property of heat equation to linear thermoelasticity and other theories. Q. Appl. Math. 40, 319–330 (1982)

Diening, L., Harjulehto, P., Hasto, P., Ruzicka, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics. Springer, Heidelberg (2011)

Ferreira, R., de Pablo, A., Perez-LLanos, M., Rossi, J.D.: Critical exponents for a semilinear parabolic equation with variable reaction. R. Soc. Edinb. Sect. A Math. 142, 1027–1042 (2012)

Fujita, H.: On the blowing up of solutions of the Cauchy problem for $$u_{t}=\Delta u+u^{1+\alpha }$$. J. Fac. Sci. Univ. Tokyo Sect. I(13), 109–124 (1966)

Furter, J., Grinfield, M.: Local vs. nonlocal interactions in populations dynamics. J. Math. Biol. 27, 65–80 (1989)

Friedman, A.: Partial Differential Equations of Parabolic Type. Prentice Hall, Englewood Cliffs (1964)

Hu, B.: Blow-Up Theories for Semilinear Parabolic Equations. Springer, Berlin (2011)

Ishii, H.: Asymptotic stability and blowing up of solutions of some nonlinear equations. J. Differ. Equ. 26, 291–319 (1977)

Ladyzenskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and Quasi-Linear Equations of Parabolic Type. American Mathematical Society, Providence (1968)

Levine, S., Chen, Y.M., Stanich, J.: Image Restoration Via Nonstandard Diffusion. Department of Mathematics and Computer Science, Duquesne University, Pittsburgh (2004)

Li, F.J., Liu, B.C.: Asymptotic analysis for blow-up solutions in parabolic equations involving variable exponents. Appl. Anal. 92, 651–664 (2013)

Liu, B.C., Yang, J.: Blow-up properties in parabolic problems with anisotropic nonstandard growth conditions. Z. Angew. Math. Phys. 67, 1–26 (2016)

Liu, B.C., Xin, Q.N., Dong, M.Z.: Blow-up analyses in parabolic equations with anisotropic nonstandard damping source. J. Math. Anal. Appl. 458, 242–262 (2018)

Mailly, G.P.: Blow up for reaction diffusion equations under dynamical boundary conditions. Commun. Partial Differ. Equ. 28, 223–247 (2003)

Payne, L.E., Sattinger, D.H.: Saddle points and instability of nonlinear hyperbolic equations. Isr. J. Math. 22, 273–303 (1975)

Pinasco, J.P.: Blow-up for parabolic and hyperbolic problems with variable exponents. Nonlinear Anal. 71, 1094–1099 (2009)

Quittner, P., Souplet, P.: Superlinear parabolic problems. blow-up, global existence and steady states. In: Birkhäuser Advanced Texts Basler Lehrbücher book series (BAT) (2019)

Rajagopal, K., Ruzicka, M.: Mathematical modelling of electro-rheological fluids. Contin. Mech. Thermodyn. 13, 59–78 (2001)

Souplet, P.: Uniform blow-up profiles and boundary behavior for diffusion equations with nonlocal nonlinear source. J. Differ. Equ. 153, 374–406 (1999)

Tan, Z.: The reaction diffusion equations with special diffusion processes (Chinese). Chin. Ann. Math. 22A, 597–606 (2001)

Tan, Z.: Non-Newton filtration equation with special medium void. Acta Math. Sci. 24B, 118–128 (2004)

Tsutsumi, M.: Existence and nonexistence of global solutions for nonlinear parabolic equations. Publ. Res. Inst. Math. Sci. 73, 211–229 (1972)

Wang, Y.: The existence of global solution and the blowup problem for some p-Laplace heat equations. Acta Math. Sci. 27B, 274–282 (2007)

Wu, X.L., Guo, B., Gao, W.J.: Blow-up of solutions for a semilinear parabolic equation involving variable source and positive initial energy. Appl. Math. Lett. 26, 539–543 (2013)

Xu, X.J., Ye, Z.: Life span of solutions with large initial data for a class of coupled parabolic systems. Z. Angew. Math. Phys. 64, 705–717 (2013)

Zhou, J.: A multi-dimension blow-up problem to a porous medium diffusion equation with special medium void. Appl. Math. Lett. 30, 6–11 (2014)