Cauchy problem for the non-Newtonian polytropic filtration equation with a localized reaction

References

• Bebernes J, Eberly D. Mathematical problems from combustion theory. New York (NY): Springer-Verlag; 1989.
   Google Scholar
• Pao CV. Nonlinear parabolic and elliptic equations. New York (NY): Plenum Press; 1992.
   Google Scholar
• Wu ZQ, Zhao JN, Yin JX, Li HL. Nonlinear diffusion equations. River Edge (NJ): World Scientific Publishing Co., Inc.; 2001.
   Google Scholar
• Zhao JN. On the Cauchy problem and initial traces for the evolution p-Laplacian equations with strongly nonlinear sources. J. Differ. Equations. 1995;121:329–383.
   Web of Science ®Google Scholar
• Fujita H. On the blowing up of solutions of the Cauchy problem for ut = ∆u + uα+1. J. Fac. Sci. Univ. Tokyo Sec. A. 1966;16:105–113.
   Google Scholar
• Deng K, Levine HA. The role of critical exponents in blow-up theorems: the sequel. J. Math. Anal. Appl. 2000;243:85–126.
   Web of Science ®Google Scholar
• Levine HA. The role of critical exponents in blow-up theorems. SIAM Rev. 1990;32:262–288.
   Web of Science ®Google Scholar
• Afanas’eva NV, Tedeev AF. Fujita-type theorems for quasilinear parabolic equations in the case of slowly decaying initial data. Mat. Sb. 2004;195:3–22 (in Russian). Translation in Sb. Math. 2004;195:459–478.
   Google Scholar
• Galaktionov VA. Blow-up for quasilinear heat equations with critical Fujita’s exponents. Proc. Roy. Soc. Edinburgh. 1994;124:517–525.
   Google Scholar
• Qi YW. The critical exponents of degenerate parabolic equations. Sci. China. Ser. A. 1994;38:1153–1162.
   Google Scholar
• Qi YW, Wang MX. Critical exponents of quasilinear parabolic equations. J. Math. Anal. Appl. 2002;267:264–280.
   Web of Science ®Google Scholar
• Winkler M. A critical exponent in a degenerate parabolic equation. Math. Meth. Appl. Sci. 2002;25:911–925.
   Web of Science ®Google Scholar
• Galaktionov VA, Levine HA. A general approach to critical Fujita exponents and systems. Nonlinear Anal. 1998;34:1005–1027.
   Web of Science ®Google Scholar
• Liu CC. Critical exponent for a quasilinear parabolic equation with inhomogeneous density in a cone. Monatsh. Math. 2012;165:237–249.
   Web of Science ®Google Scholar
• Qi YW. The critical exponents of parabolic equations and blow-up in RN. Proc. Roy. Soc. Edinburgh Sect. A. 1998;128:123–136.
   Google Scholar
• Wang CP, Zheng SN. Critical Fujita exponents of degenerate and singular parabolic equations. Proc. R. Soc. Edinb. Sect. A. 2006;136:415–430.
   Google Scholar
• Liu XF, Wang MX. The critical exponent of doubly singular parabolic equations. J. Math. Anal. Appl. 2001;257:170–188.
   Web of Science ®Google Scholar
• Yang CX, Zheng SN, Zhou SS. Critical exponents in a degenerate parabolic equation with weighted source. Appl. Anal. 2013;92:814–830.
   Web of Science ®Google Scholar
• Guo JS, Guo YJ. On a fast diffusion equation with source. Tohoku Math. J. 2001;53:571–579.
   Web of Science ®Google Scholar
• Li ZP, Du WJ. Life span and secondary critical exponent for degenerate and singular parabolic equations. Annali di Matematica. 2012; doi: 10.1007/s10231-012-0287-5.
   Web of Science ®Google Scholar
• Martynenko AV, Tedeev AF, Shramenko VN. The Cauchy problem for a degenerate parabolic equation with inhomogeneous density and source in the class of slowly dacaying initial data. Izvestiya Math. 2012;76:563–580.
   Web of Science ®Google Scholar
• Mu CL, Li YH, Wang Y. Life span and a new critical exponent for a quasilinear degenerate parabolic equation with slow decay initial values. Nonlinear Anal. RWA. 2010;11:198–206.
   Web of Science ®Google Scholar
• Mu CL, Zeng R, Zhou SM. Life span and a new critical exponent for a doubly degenerate parabolic equation with slow decay initial values. J. Math. Anal. Appl. 2011;384:181–191.
   Web of Science ®Google Scholar
• Yang JG, Yang CX, Zheng SN. Second critical exponent for evolution p-Laplacian equation with weighted source. Math. Comput. Model. 2012;56:247–256.
   Web of Science ®Google Scholar
• Bimpong-Bota K, Ortoleva P, Ross J. Far from equilibrium phenomena at local sites of reaction. J. Chem. Phys. 1974;60:3124–3133.
   Web of Science ®Google Scholar
• Bai XL, Zhou SS, Zheng SN. Cauchy problem for fast diffusion equation with localized reaction. Nonlinear Anal. 2011;74:2508–2514.
   Web of Science ®Google Scholar
• Ferreira R, de Pablo A, Vazquez JL. Classification of blow-up with nonlinear diffusion and localized reaction. J. Differ. Equations. 2006;231:195–211.
   Web of Science ®Google Scholar
• Liang ZL. On the critical exponents for a porous medium equation with a localized reaction in high dimensions. Comm. Pure Appl. Anal. 2012;11:649–658.
   Web of Science ®Google Scholar
• Liang ZL. Critical exponent for the evolution p-Laplacian eqaution with a localized reaction. Indian J. Pure Appl. Math. 2012;43:535–558.
   Web of Science ®Google Scholar
• Pinsky RG. Existence and nonexistence of global solutions for ut = ∆u + a(x)up in Rd. J. Differ. Equations. 1997;133:152–177.
   Web of Science ®Google Scholar
• Wang ZJ, Yin JX, Wang CP. Critical exponents of the non-Newtonian polytropic filtration equation with nonlinear boundary condition. Appl. Math. Lett. 2007;20:142–147.
   Web of Science ®Google Scholar
• Mu CL, Zheng P, Liu DM. Localization of solutions to a doubly degenerate parabolic equation with a strongly nonlinear source. Commun. Contemp. Math. 2012;14. Article ID 1250018:1–18.
   Web of Science ®Google Scholar
• Liang ZL, Zhao JN. Localization for the evolution p-Laplacian equation with strongly nonlinear source term. J. Differ. Equations. 2009;246:391–407.
   Web of Science ®Google Scholar
• Zhao JN. Existence of nonexistence of solutions for ut = div(|∇u|p−2∇u) + f(∇u, u, x, t). J. Math. Anal. Appl. 1993;172:130–146.
   Web of Science ®Google Scholar
• Samarskii A, Galaktionov V, Kurdyumov V, Mikhailov A. Blow-up in quasilinear parabolic equations. Berlin: Walter de Gruyter; 1995.
   Google Scholar
• Serrin J, Zou H. Cauchy-Liouville and universal boundedness theorems for quasilinear elliptic equations and inequalities. Acta. Math. 2002;189:79–142.
   Web of Science ®Google Scholar
• Galaktionov VA, Levine HA. On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary. Israel J. Math. 1996;94:125–146.
   Web of Science ®Google Scholar
• Stan D, Vazquez JL. Asymptotic behaviour of the doubly nonlinear diffusion equation ut = ∆pum on bounded domains. Nonlinear Anal. 2013;77:1–32.
   Web of Science ®Google Scholar
• Levine HA, Sacks P. Some existence and nonexistence theorems for solutions of degenerate parabolic equations. J. Differ. Equations. 1984;52:135–161.
   Web of Science ®Google Scholar
• Giga Y, Kohn RV. Characterizing blowup using similarity variables. Indiana Univ. Math. J. 1987;36:1–40.
   Web of Science ®Google Scholar