Abstract
In this paper, we establish some new Liouville-type results for solutions of nonlinear degenerate parabolic system of inequalities. Nonexistence of nontrivial global solutions to initial-value problems is studied by using scaling transformations and test functions.
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F. Antonelli, E. Barucci, and A. Pascucci, “A comparison result for FBSDE with applications to decisions theory,” Math. Methods Oper. Res. 54, 407–423 (2001).
F. Antonelli and A. Pascucci, “On the viscosity solutions of a stochastic differential utility problem,” J. Differ. Equations 186, 69–87 (2002).
H. Wu, “Blow-up and nonexistence of solutions of some semilinear degenerate parabolic equations,” Bound. Value Probl. 157, 1–12 (2015).
P. Polácik, P. Quittner, and Ph. Souplet, “Singularity and decay estimates in superlinear problems via Liouville-type theorems, Part I: Elliptic equations and systems,” Duke Math. J. 139, 555–579 (2007).
P. Polácik, P. Quittner, and Ph. Souplet, “Singularity and decay estimates in superlinear problems via Liouville-type theorems, Part II: Parabolic equations,” Indiana Univ. Math. J. 56, 879–908 (2007).
J. Földes, “Liouville theorems, a priori estimates, and blow-up rates for solutions of indefinite superlinear parabolic problems,” Czech. Math. J. 136 (61), 169–198 (2011).
K. Hayakawa, “On nonexistence of global solutions of some semilinear parabolic differential equations,” Proc. Japan Acad. Ser. A 49, 503–505 (1973).
B. Gidas and J. Spruck, “Global and local behavior of positive solutions of nonlinear elliptic equations,” Comm. Pure Appl. Math. 34(4), 525–598 (1981).
M. Escobedo and M. A. Herrero, “Boundedness and blow up for a semilinear reaction-diffusion system,” J. Differ. Equations 89, 176–202 (1991).
A. G. Kartsatos and V. V. Kurta, “On a Liouville-type theorem and the Fujita blow-up phenomenon,” Proc. Amer. Math. Soc. 132, 807–813 (2004).
M. Chipot and F. B. Weissler, “Some blowup results for a nonlinear parabolic equation with a gradient term,” SIAM J. Math. Anal. 20, 886–907 (1989).
E. Mitidieri and S. I. Pokhozhaev, “A priori estimates and blow-up of solutions to nonlinear partial differential equations and inequalities,” Tr. Mat. Inst. Steklova 234, 3–383 (2001).
P. Polácik and P. Quittner, “Liouville-type theorems and complete blow-up for indefinite superlinear parabolic equations,” in Progress in Nonlinear Differential Equations and Their Applications (Birkhäuser, Basel, 2005), Vol. 64, pp. 391–402.
F. Sh. Li and J. L. Li, “Global existence and blow-up phenomena for nonlinear divergence form parabolic equations with inhomogeneous Neumann boundary conditions,” J. Math. Anal. Appl. 385, 1005–1014 (2012).
Y. Du and S. Li, “Nonlinear Liouville theorems and a priori estimates for indefinite superlinear elliptic equations,” Adv. Differential Equations 10, 841–860 (2005).
S. N. Armstrong and B. Sirakov, “Nonexistence of positive supersolutions of elliptic equations via the maximum principle,” Comm. Partial Differential Equations 36, 2011–2047 (2011).
G. Caristi, “Existence and nonexistence of global solutions of degenerate and singular parabolic systems,” Abstr. Appl. Anal. 5, 265–284 (2000).
L. H. Min, “Liouville-type theorem for higher-order Hardy-Hénon system of inequalities,” Math. Inequal. Appl. 17 (4), 1427–1439 (2014).
P. Quittner and Ph. Souplet, Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States, in Birkhäuser Advanced Texts (Birkhäuser Verlag, Basel, 2007).
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Wu, H. Liouville-Type Theorem for a Nonlinear Degenerate Parabolic System of Inequalities. Math Notes 103, 155–163 (2018). https://doi.org/10.1134/S0001434618010170
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DOI: https://doi.org/10.1134/S0001434618010170