# 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (2007). Google Scholar [25] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness,, Academic Press, (1975). Google Scholar [26] S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations,, Osaka J. Math., 12 (1975), 45. Google Scholar [27] X. Wang, On the Cauchy problem for reaction-diffusion equations,, Trans. Amer. Math. Soc., 337 (1993), 549. 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. doi: 10.1007/BF02761845. Google Scholar [29] W. P. Ziemer, Weakly Differentiable Functions,, Springer-Verlag, (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar

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##### References:
 [1] L. Amour and M. Ben-Artzi, Global existence and decay for viscous Hamilton-Jacobi equations,, Nonlinear Anal., 31 (1998), 621. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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. 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, (2007). Google Scholar [25] M. Reed and B. Simon, Methods of Modern Mathematical Physics. II. Fourier Analysis, Self-Adjointness,, Academic Press, (1975). Google Scholar [26] S. Sugitani, On nonexistence of global solutions for some nonlinear integral equations,, Osaka J. Math., 12 (1975), 45. Google Scholar [27] X. Wang, On the Cauchy problem for reaction-diffusion equations,, Trans. Amer. Math. Soc., 337 (1993), 549. 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. doi: 10.1007/BF02761845. Google Scholar [29] W. P. Ziemer, Weakly Differentiable Functions,, Springer-Verlag, (1989). doi: 10.1007/978-1-4612-1015-3. Google Scholar
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