# American Institute of Mathematical Sciences

August  2014, 7(4): 767-783. doi: 10.3934/dcdss.2014.7.767

## Global solutions for a nonlinear integral equation with a generalized heat kernel

 1 Mathematical Institute, Tohoku University, Aoba, Sendai 980-8578, Japan 2 Department of Mathematical Sciences, Osaka Prefecture University, Sakai 599-8531, Japan

Received  September 2013 Published  February 2014

We study the existence and the large time behavior of global-in-time solutions of a nonlinear integral equation with a generalized heat kernel \begin{eqnarray*} & & u(x,t)=\int_{{\mathbb R}^N}G(x-y,t)\varphi(y)dy\\ & & \qquad\quad +\int_0^t\int_{{\mathbb R}^N}G(x-y,t-s)F(y,s,u(y,s),\dots,\nabla^\ell u(y,s))dyds, \end{eqnarray*} where $\varphi\in W^{\ell,\infty}({\mathbb R}^N)$ and $\ell\in\{0,1,\dots\}$. The arguments of this paper are applicable to the Cauchy problem for various nonlinear parabolic equations such as fractional semilinear parabolic equations, higher order semilinear parabolic equations and viscous Hamilton-Jacobi equations.
Citation: Kazuhiro Ishige, Tatsuki Kawakami, Kanako Kobayashi. Global solutions for a nonlinear integral equation with a generalized heat kernel. Discrete & Continuous Dynamical Systems - S, 2014, 7 (4) : 767-783. doi: 10.3934/dcdss.2014.7.767
##### References:
 [1] L. Amour and M. Ben-Artzi, Global existence and decay for viscous Hamilton-Jacobi equations, Nonlinear Anal., 31 (1998), 621-628. doi: 10.1016/S0362-546X(97)00427-6.  Google Scholar [2] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.  Google Scholar [3] P. Biler, T. Funaki and W. A. Woyczynski, Fractal Burgers equations, J. Differential Equations, 148 (1998), 9-46. doi: 10.1006/jdeq.1998.3458.  Google Scholar [4] P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. Math., 59 (1999), 845-869. doi: 10.1137/S0036139996313447.  Google Scholar [5] S. Benachour, G. Karch and P. Laurençot, Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations, J. Math. Pures Appl., 83 (2004), 1275-1308. doi: 10.1016/j.matpur.2004.03.002.  Google Scholar [6] G. Caristi and E. Mitidieri, Existence and nonexistence of global solutions of higher-order parabolic problems with slow decay initial data, J. Math. Anal. Appl., 279 (2003), 710-722. doi: 10.1016/S0022-247X(03)00062-3.  Google Scholar [7] S. Cui, Local and global existence of solutions to semilinear parabolic initial value problems, Nonlinear Anal., 43 (2001), 293-323. doi: 10.1016/S0362-546X(99)00195-9.  Google Scholar [8] Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S. I. Pohozaev, On the necessary conditions of global existence to a quasilinear inequality in the half-space, C. R. Math. Acad. Sci. Paris, 330 (2000), 93-98. doi: 10.1016/S0764-4442(00)00124-5.  Google Scholar [9] M. Fila, K. Ishige and T. Kawakami, Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition, Commun. Pure Appl. Anal., 11 (2012), 1285-1301. doi: 10.3934/cpaa.2012.11.1285.  Google Scholar [10] A. Fino and G. Karch, Decay of mass for nonlinear equation with fractional Laplacian, Monatsh. Math., 160 (2010), 375-384. doi: 10.1007/s00605-009-0093-3.  Google Scholar [11] H. Fujita, 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 (1966), 109-124.  Google Scholar [12] V. A. Galaktionov and S. I. Pohozaev, Existence and blow-up for higher-order semilinear parabolic equations: majorizing order-preserving operators, Indiana Univ. Math. J., 51 (2002), 1321-1338. doi: 10.1512/iumj.2002.51.2131.  Google Scholar [13] L. Grafakos, Classical Fourier Analysis, Springer-Verlag, 2008.  Google Scholar [14] F. Gazzola and H.-C. Grunau, Global solutions for superlinear parabolic equations involving the biharmonic operator for initial data with optimal slow decay, Calc. Var. Partial Differential Equations, 30 (2007), 389-415. doi: 10.1007/s00526-007-0096-7.  Google Scholar [15] K. Hayakawa, On the nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad., 49 (1973), 503-525. doi: 10.3792/pja/1195519254.  Google Scholar [16] K. Ishige, M. Ishiwata and T. Kawakami, The decay of the solutions for the heat equation with a potential, Indiana Univ. Math. J., 58 (2009), 2673-2708. doi: 10.1512/iumj.2009.58.3771.  Google Scholar [17] K. Ishige and T. Kawakami, Asymptotic expansions of solutions of the Cauchy problem for nonlinear parabolic equations, J. Anal. Math., 121 (2013), 317-351. doi: 10.1007/s11854-013-0038-6.  Google Scholar [18] K. Ishige, T. Kawakami and K. Kobayashi, Asymptotics for a nonlinear integral equation with a generalized heat kernel,, preprint, ().   Google Scholar [19] T. Kawanago, Existence and behaviour of solutions for $u_t=\Delta(u^m)+u^l$, Adv. Math. Sci. Appl., 7 (1997), 367-400.  Google Scholar [20] K. Kobayashi, T. Sirao and H. Tanaka, On the glowing up problem for semilinear heat equations, J. Math. Soc. Japan, 29 (1977), 407-424. doi: 10.2969/jmsj/02930407.  Google Scholar [21] P. Laurençot and P. Souplet, On the growth of mass for a viscous Hamilton-Jacobi equation, J. Anal. Math., 89 (2003), 367-383. doi: 10.1007/BF02893088.  Google Scholar [22] T. Y. Lee and W. M. Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Soc., 333 (1992), 365-378. doi: 10.1090/S0002-9947-1992-1057781-6.  Google Scholar [23] G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418. doi: 10.1016/0362-546X(85)90001-X.  Google Scholar [24] P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Basel, 2007.  Google Scholar [25] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York and London, 1975.  Google Scholar [26] S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math., 12 (1975), 45-51.  Google Scholar [27] X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590. doi: 10.1090/S0002-9947-1993-1153016-5.  Google Scholar [28] F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845.  Google Scholar [29] W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar

show all references

##### References:
 [1] L. Amour and M. Ben-Artzi, Global existence and decay for viscous Hamilton-Jacobi equations, Nonlinear Anal., 31 (1998), 621-628. doi: 10.1016/S0362-546X(97)00427-6.  Google Scholar [2] D. G. Aronson and H. F. Weinberger, Multidimensional nonlinear diffusion arising in population genetics, Adv. in Math., 30 (1978), 33-76. doi: 10.1016/0001-8708(78)90130-5.  Google Scholar [3] P. Biler, T. Funaki and W. A. Woyczynski, Fractal Burgers equations, J. Differential Equations, 148 (1998), 9-46. doi: 10.1006/jdeq.1998.3458.  Google Scholar [4] P. Biler and W. A. Woyczyński, Global and exploding solutions for nonlocal quadratic evolution problems, SIAM J. Appl. Math., 59 (1999), 845-869. doi: 10.1137/S0036139996313447.  Google Scholar [5] S. Benachour, G. Karch and P. Laurençot, Asymptotic profiles of solutions to viscous Hamilton-Jacobi equations, J. Math. Pures Appl., 83 (2004), 1275-1308. doi: 10.1016/j.matpur.2004.03.002.  Google Scholar [6] G. Caristi and E. Mitidieri, Existence and nonexistence of global solutions of higher-order parabolic problems with slow decay initial data, J. Math. Anal. Appl., 279 (2003), 710-722. doi: 10.1016/S0022-247X(03)00062-3.  Google Scholar [7] S. Cui, Local and global existence of solutions to semilinear parabolic initial value problems, Nonlinear Anal., 43 (2001), 293-323. doi: 10.1016/S0362-546X(99)00195-9.  Google Scholar [8] Yu. V. Egorov, V. A. Galaktionov, V. A. Kondratiev and S. I. Pohozaev, On the necessary conditions of global existence to a quasilinear inequality in the half-space, C. R. Math. Acad. Sci. Paris, 330 (2000), 93-98. doi: 10.1016/S0764-4442(00)00124-5.  Google Scholar [9] M. Fila, K. Ishige and T. Kawakami, Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition, Commun. Pure Appl. Anal., 11 (2012), 1285-1301. doi: 10.3934/cpaa.2012.11.1285.  Google Scholar [10] A. Fino and G. Karch, Decay of mass for nonlinear equation with fractional Laplacian, Monatsh. Math., 160 (2010), 375-384. doi: 10.1007/s00605-009-0093-3.  Google Scholar [11] H. Fujita, 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 (1966), 109-124.  Google Scholar [12] V. A. Galaktionov and S. I. Pohozaev, Existence and blow-up for higher-order semilinear parabolic equations: majorizing order-preserving operators, Indiana Univ. Math. J., 51 (2002), 1321-1338. doi: 10.1512/iumj.2002.51.2131.  Google Scholar [13] L. Grafakos, Classical Fourier Analysis, Springer-Verlag, 2008.  Google Scholar [14] F. Gazzola and H.-C. Grunau, Global solutions for superlinear parabolic equations involving the biharmonic operator for initial data with optimal slow decay, Calc. Var. Partial Differential Equations, 30 (2007), 389-415. doi: 10.1007/s00526-007-0096-7.  Google Scholar [15] K. Hayakawa, On the nonexistence of global solutions of some semilinear parabolic equations, Proc. Japan Acad., 49 (1973), 503-525. doi: 10.3792/pja/1195519254.  Google Scholar [16] K. Ishige, M. Ishiwata and T. Kawakami, The decay of the solutions for the heat equation with a potential, Indiana Univ. Math. J., 58 (2009), 2673-2708. doi: 10.1512/iumj.2009.58.3771.  Google Scholar [17] K. Ishige and T. Kawakami, Asymptotic expansions of solutions of the Cauchy problem for nonlinear parabolic equations, J. Anal. Math., 121 (2013), 317-351. doi: 10.1007/s11854-013-0038-6.  Google Scholar [18] K. Ishige, T. Kawakami and K. Kobayashi, Asymptotics for a nonlinear integral equation with a generalized heat kernel,, preprint, ().   Google Scholar [19] T. Kawanago, Existence and behaviour of solutions for $u_t=\Delta(u^m)+u^l$, Adv. Math. Sci. Appl., 7 (1997), 367-400.  Google Scholar [20] K. Kobayashi, T. Sirao and H. Tanaka, On the glowing up problem for semilinear heat equations, J. Math. Soc. Japan, 29 (1977), 407-424. doi: 10.2969/jmsj/02930407.  Google Scholar [21] P. Laurençot and P. Souplet, On the growth of mass for a viscous Hamilton-Jacobi equation, J. Anal. Math., 89 (2003), 367-383. doi: 10.1007/BF02893088.  Google Scholar [22] T. Y. Lee and W. M. Ni, Global existence, large time behavior and life span of solutions of a semilinear parabolic Cauchy problem, Trans. Amer. Math. Soc., 333 (1992), 365-378. doi: 10.1090/S0002-9947-1992-1057781-6.  Google Scholar [23] G. Ponce, Global existence of small solutions to a class of nonlinear evolution equations, Nonlinear Anal., 9 (1985), 399-418. doi: 10.1016/0362-546X(85)90001-X.  Google Scholar [24] P. Quittner and P. Souplet, Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States, Birkhäuser Advanced Texts, Basel, 2007.  Google Scholar [25] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness, Academic Press, New York and London, 1975.  Google Scholar [26] S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations, Osaka J. Math., 12 (1975), 45-51.  Google Scholar [27] X. Wang, On the Cauchy problem for reaction-diffusion equations, Trans. Amer. Math. Soc., 337 (1993), 549-590. doi: 10.1090/S0002-9947-1993-1153016-5.  Google Scholar [28] F. B. Weissler, Existence and nonexistence of global solutions for a semilinear heat equation, Israel J. Math., 38 (1981), 29-40. doi: 10.1007/BF02761845.  Google Scholar [29] W. P. Ziemer, Weakly Differentiable Functions, Springer-Verlag, New York, 1989. doi: 10.1007/978-1-4612-1015-3.  Google Scholar
 [1] Pavol Quittner. The decay of global solutions of a semilinear heat equation. Discrete & Continuous Dynamical Systems, 2008, 21 (1) : 307-318. doi: 10.3934/dcds.2008.21.307 [2] Kazuhiro Ishige, Ryuichi Sato. Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces. Discrete & Continuous Dynamical Systems, 2016, 36 (5) : 2627-2652. doi: 10.3934/dcds.2016.36.2627 [3] Kazuhiro Ishige, Tatsuki Kawakami. Asymptotic behavior of solutions for some semilinear heat equations in $R^N$. Communications on Pure & Applied Analysis, 2009, 8 (4) : 1351-1371. doi: 10.3934/cpaa.2009.8.1351 [4] Soohyun Bae. Weighted $L^\infty$ stability of positive steady states of a semilinear heat equation in $\R^n$. Discrete & Continuous Dynamical Systems, 2010, 26 (3) : 823-837. doi: 10.3934/dcds.2010.26.823 [5] Kazuhiro Ishige, Michinori Ishiwata. Global solutions for a semilinear heat equation in the exterior domain of a compact set. Discrete & Continuous Dynamical Systems, 2012, 32 (3) : 847-865. doi: 10.3934/dcds.2012.32.847 [6] Keisuke Matsuya, Tetsuji Tokihiro. Existence and non-existence of global solutions for a discrete semilinear heat equation. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 209-220. doi: 10.3934/dcds.2011.31.209 [7] J. Colliander, Justin Holmer, Monica Visan, Xiaoyi Zhang. Global existence and scattering for rough solutions to generalized nonlinear Schrödinger equations on $R$. Communications on Pure & Applied Analysis, 2008, 7 (3) : 467-489. doi: 10.3934/cpaa.2008.7.467 [8] Mingyou Zhang, Qingsong Zhao, Yu Liu, Wenke Li. Finite time blow-up and global existence of solutions for semilinear parabolic equations with nonlinear dynamical boundary condition. Electronic Research Archive, 2020, 28 (1) : 369-381. doi: 10.3934/era.2020021 [9] Chien-Hong Cho, Marcus Wunsch. Global weak solutions to the generalized Proudman-Johnson equation. Communications on Pure & Applied Analysis, 2012, 11 (4) : 1387-1396. doi: 10.3934/cpaa.2012.11.1387 [10] Zhengce Zhang, Yan Li. Global existence and gradient blowup of solutions for a semilinear parabolic equation with exponential source. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 3019-3029. doi: 10.3934/dcdsb.2014.19.3019 [11] Francesca De Marchis, Isabella Ianni. Blow up of solutions of semilinear heat equations in non radial domains of $\mathbb{R}^2$. Discrete & Continuous Dynamical Systems, 2015, 35 (3) : 891-907. doi: 10.3934/dcds.2015.35.891 [12] Ludovic Dan Lemle. $L^1(R^d,dx)$-uniqueness of weak solutions for the Fokker-Planck equation associated with a class of Dirichlet operators. Electronic Research Announcements, 2008, 15: 65-70. doi: 10.3934/era.2008.15.65 [13] Thierry Horsin, Peter I. Kogut, Olivier Wilk. Optimal $L^2$-control problem in coefficients for a linear elliptic equation. II. Approximation of solutions and optimality conditions. Mathematical Control & Related Fields, 2016, 6 (4) : 595-628. doi: 10.3934/mcrf.2016017 [14] P. R. Zingano. Asymptotic behavior of the $L^1$ norm of solutions to nonlinear parabolic equations. Communications on Pure & Applied Analysis, 2004, 3 (1) : 151-159. doi: 10.3934/cpaa.2004.3.151 [15] Peiying Chen. Existence and uniqueness of weak solutions for a class of nonlinear parabolic equations. Electronic Research Announcements, 2017, 24: 38-52. doi: 10.3934/era.2017.24.005 [16] Fabio Punzo. Support properties of solutions to nonlinear parabolic equations with variable density in the hyperbolic space. Discrete & Continuous Dynamical Systems - S, 2012, 5 (3) : 657-670. doi: 10.3934/dcdss.2012.5.657 [17] Xiumei Deng, Jun Zhou. Global existence and blow-up of solutions to a semilinear heat equation with singular potential and logarithmic nonlinearity. Communications on Pure & Applied Analysis, 2020, 19 (2) : 923-939. doi: 10.3934/cpaa.2020042 [18] Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136 [19] Thierry Cazenave, Flávio Dickstein, Fred B. Weissler. Universal solutions of the heat equation on $\mathbb R^N$. Discrete & Continuous Dynamical Systems, 2003, 9 (5) : 1105-1132. doi: 10.3934/dcds.2003.9.1105 [20] Hideo Kubo, Kotaro Tsugawa. Global solutions and self-similar solutions of the coupled system of semilinear wave equations in three space dimensions. Discrete & Continuous Dynamical Systems, 2003, 9 (2) : 471-482. doi: 10.3934/dcds.2003.9.471

2019 Impact Factor: 1.233