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

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.  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

[1]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

[2]

Cheng He, Changzheng Qu. Global weak solutions for the two-component Novikov equation. Electronic Research Archive, 2020, 28 (4) : 1545-1562. doi: 10.3934/era.2020081

[3]

Bo Chen, Youde Wang. Global weak solutions for Landau-Lifshitz flows and heat flows associated to micromagnetic energy functional. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020268

[4]

Hirokazu Ninomiya. Entire solutions of the Allen–Cahn–Nagumo equation in a multi-dimensional space. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 395-412. doi: 10.3934/dcds.2020364

[5]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[6]

Hua Chen, Yawei Wei. Multiple solutions for nonlinear cone degenerate elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020272

[7]

Thierry Cazenave, Ivan Naumkin. Local smooth solutions of the nonlinear Klein-gordon equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020448

[8]

Touria Karite, Ali Boutoulout. Global and regional constrained controllability for distributed parabolic linear systems: RHum approach. Numerical Algebra, Control & Optimization, 2020  doi: 10.3934/naco.2020055

[9]

Reza Chaharpashlou, Abdon Atangana, Reza Saadati. On the fuzzy stability results for fractional stochastic Volterra integral equation. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020432

[10]

Mathew Gluck. Classification of solutions to a system of $ n^{\rm th} $ order equations on $ \mathbb R^n $. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5413-5436. doi: 10.3934/cpaa.2020246

[11]

Aihua Fan, Jörg Schmeling, Weixiao Shen. $ L^\infty $-estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 297-327. doi: 10.3934/dcds.2020363

[12]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020276

[13]

Bahaaeldin Abdalla, Thabet Abdeljawad. Oscillation criteria for kernel function dependent fractional dynamic equations. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020443

[14]

Anna Abbatiello, Eduard Feireisl, Antoní Novotný. Generalized solutions to models of compressible viscous fluids. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 1-28. doi: 10.3934/dcds.2020345

[15]

Jianhua Huang, Yanbin Tang, Ming Wang. Singular support of the global attractor for a damped BBM equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020345

[16]

Serge Dumont, Olivier Goubet, Youcef Mammeri. Decay of solutions to one dimensional nonlinear Schrödinger equations with white noise dispersion. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020456

[17]

Huiying Fan, Tao Ma. Parabolic equations involving Laguerre operators and weighted mixed-norm estimates. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5487-5508. doi: 10.3934/cpaa.2020249

[18]

Jun Zhou. Lifespan of solutions to a fourth order parabolic PDE involving the Hessian modeling epitaxial growth. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5581-5590. doi: 10.3934/cpaa.2020252

[19]

Gunther Uhlmann, Jian Zhai. Inverse problems for nonlinear hyperbolic equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 455-469. doi: 10.3934/dcds.2020380

[20]

Leilei Wei, Yinnian He. A fully discrete local discontinuous Galerkin method with the generalized numerical flux to solve the tempered fractional reaction-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020319

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (80)
  • HTML views (0)
  • Cited by (9)

[Back to Top]