RESEARCH ARTICLE
Global existence of solutions for a p-Laplacian equation with nonlocal Fisher–KPP type reaction terms
Xueyan Tao,
Zhong Bo Fang,
Xueyan Tao
School of Mathematical Sciences, Peking University, Beijing, 100871 P. R. China
Search for more papers by this authorCorresponding Author
Zhong Bo Fang
School of Mathematical Sciences, Ocean University of China, Qingdao, 266100 P. R. China
Correspondence
Zhong Bo Fang, School of Mathematical Sciences, Ocean University of China, 266100 Qingdao, China.
Email: [email protected]
Communicated by: Y. Xu
Search for more papers by this authorXueyan Tao,
Zhong Bo Fang,
Xueyan Tao
School of Mathematical Sciences, Peking University, Beijing, 100871 P. R. China
Search for more papers by this authorCorresponding Author
Zhong Bo Fang
School of Mathematical Sciences, Ocean University of China, Qingdao, 266100 P. R. China
Correspondence
Zhong Bo Fang, School of Mathematical Sciences, Ocean University of China, 266100 Qingdao, China.
Email: [email protected]
Communicated by: Y. Xu
Search for more papers by this authorAbstract
This work is concerned with the Neumann initial boundary value problem and Cauchy problem of a parabolic p-Laplacian equation with nonlocal Fisher–KPP type reaction terms. We establish a uniform boundedness and global existence of solutions to the equation by applying the method of multipliers and modified Moser's iteration technique for some ranges of parameters. The ranges of parameters have similar structure to that of the classical critical Fujita exponent.
CONFLICT OF INTEREST
This work does not have any conflicts of interest.
REFERENCES
- 1Pao CV. Nonlinear Parabolic and Elliptic Equations. New York:Plenum Press; 1992.
- 2DiBenedetto E. Degenerate Parabolic Equations. New York:Springer; 1993.
10.1007/978-1-4612-0895-2 Google Scholar
- 3Samarskii AA, Galaktionov VA, Kurdyumov SP, Mikhailov AP. Blow-Up in Quasilinear Parabolic Equations. Berlin:Walter de Gruyter; 1995.
10.1515/9783110889864 Google Scholar
- 4Wu ZQ, Zhao JN, Yin JX, Li HL. Nonlinear Diffusion Equations. River Edge:World Scientific; 2001.
10.1142/4782 Google Scholar
- 5Quittner P, Souplet Ph. Superlinear Parabolic Problems. Blow-up, Global Existence and Steady States. Birkhäuser Advanced Texts:Basel; 2007.
- 6Hu B. Blow-up Theories for Semilinear Parabolic Equations. Berlin:Springer; 2011.
- 7Galaktionov VA, Vázquez JL. The problem of blow up in nonlinear parabolic equations. Discrete Contin Dyn Syst. 2002; 8: 399-433.
- 8Levine HA. The role of critical exponents in blow-up theorems. SIAM Rev. 1990; 32: 262-288.
- 9Bebernes JW, Bressan A. Thermal behaviour for a confined reactive gas. J Differ Equa. 1982; 44: 118-133.
- 10Bebernes JW, Ely R. Comparison techniques and the method of lines for a parabolic functional equation Rocky Mountain. J Math. 1982; 12: 723-733.
- 11Pao CV. Blowing-up of solution for a nonlocal reaction-diffusion problem in combustion theory. J Math Anal Appl. 1992; 166: 591-600.
- 12Liu Q, Chen Y, Lu S. Uniform blow-up profiles for nonlinear and nonlocal reaction-diffusion equations. Nonlinear Anal. 2009; 71: 1572-1583.
- 13Wang M, Wang Y. Properties of positive solutions for non-local reaction-diffusion problems. Math Methods Appl Sci. 1996; 19: 1141-1156.
- 14Budd C, Dold B, Stewart A. Blowup in a partial differential equation with conserved first integral. SIAM J Appl Math. 1993; 53: 718-742.
- 15Hu B, Yin HM. Semilinear parabolic equations with prescribed energy. Rend Circ Mat Palermo. 1995; 44: 479-505.
- 16Niculescu CP, Rovenţa I. Large solutions for semilinear parabolic equations involving some special classes of nonlinearities. Discrete Dyn Nat Soc. 2010; 2010. Article ID 491023 11 pages.
- 17Gao WJ, Han YZ. Blow-up of a nonlocal semilinear parabolic equation with positive initial energy. Appl Math Lett. 2011; 24: 784-788.
- 18Dong Z, Zhou J. Global existence and finite time blow-up for a class of thin-film equation. Z Angew Math Phys. 2017; 68: 68-89.
- 19Anguiano M, Kloeden PE, Lorenz T. Asymptotic behaviour of nonlocal reaction-diffusion equations. Nonlinear Anal. 2010; 73: 3044-3057.
- 20Bian S, Chen L. A nonlocal reaction diffusion equation and its relation with Fujita exponent. J Math Anal Appl. 2016; 444(2): 1479-1489.
- 21Bian S. Global solutions to a nonlocal Fisher–KPP type problem. Acta Appl Math. 2017; 147: 187-195.
- 22Bian S, Chen L, Latos EA. Global existence and asymptotic behavior of solutions to a nonlocal Fisher–KPP type problem. Nonlinear Anal. 2017; 149: 165-176.
- 23Britton NF. Aggregation and the competitive exclusion principle. J Theor Biol. 1989; 136: 57-66.
- 24Niculescu CP, Rovenţa J.. Generalized convexity and the existence of finite time blow-up solutions for an evolutionary problem. Nonlinear Anal TMA. 2012; 75: 270-277.
- 25Fang ZB, Sun L, Li CJ. A note on blow-up of solutions for the nonlocal quasilinear parabolic equation with positive initial energy. Boundary Value Problems. 2013; 2013: 181.
- 26Fang ZB, Zhang JY. Influence of weight functions to a nonlocal p-Laplacian evolution equation with inner absorption and nonlocal boundary condition. J Inequal Appl. 2013; 2013: 301.
- 27Qu CY, Bai XL, Zheng SN. Blow-up versus extinction in a nonlocal p-Laplace equation with Neumann boundary conditions. J Math Anal Appl. 2014; 412: 326-333.
- 28Hamel F, Nadirashvili N, Arch Rational Mech Anal. Travelling fronts and entire solutions of the Fisher-KPP equation in . 2001; 157: 91-163.
- 29King JR, McCabe PM. On the Fisher–KPP equation with fast nonlinear diffusion. R Soc Lond Proc Ser A Math Phys Eng Sci. 2003; 459: 2529-2546.
- 30Li WT, Sun YJ, Wang ZC. Entire solutions in the Fisher-KPP equation with nonlocal dispersal. Nonlinear Anal. 2010; 11: 2302-2313.
- 31Furter J, Grinfield M. Local vs. non-local interactions in population dynamics. J Math Biol. 1989; 27: 65-80.
- 32Calsina A, Perello C, Saldana J. Non-local reaction-diffusion equations modelling predator–prey coevolution. Publ Mat. 1994; 32: 315-325.
10.5565/PUBLMAT_38294_04 Google Scholar
- 33Allegretto W, Fragnelli G, Nistri P, Papin D. Coexistence and optimal control problems for a degenerate predator–prey model. J Math Anal Appl. 2011; 378: 528-540.
- 34Lorz A, Mirrahimi S, Perthame B. Dirac mass dynamics in multidimensional nonlocal parabolic equations, Comm. Partial Differ Equa. 2011; 36(6): 1071-1098.
- 35Lorz A, Lorenzi T, Clairambault J, Escargueil A, Perthame B. Effects of space structure and combination therapies on phenotypic heterogeneity and drug resistance in solid tumors. Bull Math Biol. 2013; 77(1): 1-22.
- 36Volpert V, Vougalter V. Existence of stationary pulses for nonlocal reaction-diffusion equations. Doc Math. 2014; 19: 1141-1153.
- 37Kawohl B. On a family of torsional creep problems. J Reine Angew Math. 1990; 410: 1-22.
- 38Showalter RE, Walkington NJ. Diffusion of fluid in a fissured medium with microstructure, SIAM. J Math Anal. 1991; 22: 1702-1722.
- 39Pélissier M.-C., Reynaud L. Étude d'un modèle mathématique d'écoulement de glacier. C R Acad Sci Paris Sé,r A. 1974; 279: 531-534.
- 40Friedman A, Equations PartialDifferential. Holt. New York:Rinehart and Winston; 1969.
- 41Ladyzhenskaya OA, Solonnikov VA, Ural'tseva NN. Linear and quasilinear equations of parabolic type. Am Math Soc Providence. 1968.
- 42Kalashnikov AS. Some problems of the qualitative theory of second-order nonlinear degenerate parabolic equations. Usp Mat Nauk. 1987; 42: 135-176.
- 43Li FC, Xie CH. Global and blow-up solutions to a p-Laplacian equation with nonlocal source. Comput Math Appl. 2003; 46: 1525-1533.
- 44Galaktionov VA. Blow-up for quasilinear heat equations with critical Fujita's exponents. Proc Roy Soc Edinburgh. 1994; 124(3): 517-525.
10.1017/S0308210500028766 Google Scholar
- 45Zhao JN. Existence and nonexistence of solutions forut=div(|∇u|p−2∇u)+f(∇u,u,x,t). J Math Anal Appl. 1993; 172: 130-146.
- 46Alikakos ND. Lp bounds of solutions of reaction-diffusion equations, Commun. Partial Differ Equa. 1979; 4: 827-868.
10.1080/03605307908820113 Google Scholar
