Ir al contenido

Documat

Well-Posedness of Mild Solutions for Superdiffusion Equations with Spatial Nonlocal Operators

  • Xuan-Xuan Xi [1] ; Yong Zhou [2] ; Mimi Hou [3]
    1. [1] Xiangtan University

      Xiangtan University

      China

    2. [2] Macau University of Science and Technology

      Macau University of Science and Technology

      RAE de Macao (China)

    3. [3] Huaibei Normal University

      Huaibei Normal University

      China

    Mostrar afiliaciones +
  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 23, Nº 5, 2024
  • Idioma: inglés
  • DOI: 10.1007/s12346-024-01084-y
  • Enlaces
  • Resumen

    • In this paper, we study the well-posedness for a class of semilinear superdiffusion equations with spatial nonlocal operators. We first establish the Gagliardo–Nirenberg inequality in -Bessel potential spaces. Based on this, the well-posedness results of local and global mild solution for corresponding linear problem are obtained via apriori estimates. We also obtain the well-posedness results for the nonlinear problem under different conditions. These conclusions are mainly based on the Mihlin–Hörmander’s multiplier estimates, embedding theorem and fixed point theory.

  • Referencias bibliográficas
    • Bahouri, H., Chemin, J.-Y., Danchin, R.: Fourier Analysis and Nonlinear Partial Differential Equations. Grundlehren der mathematischen Wissenschaften,...
    • Chemin, J.-Y.: Localization in Fourier space and Navier–Stokes system. Phase space analysis of partial differential equations, vol. 1. CRM...
    • de Almeida, M.F., Precioso, J.C.P.: Existence and symmetries of solutions in Besov–Morrey spaces for a semilinear heat-wave type equation....
    • de Almeida, M.F., Viana, A.: Self-similar solutions for a superdiffusive heat equation with gradient nonlinearity. Electron. J. Differ. Equ....
    • Djida, J.-D., Fernandez, A., Area, I.: Well-posedness results for fractional semi-linear wave equations. Discrete Contin. Dyn. Syst. Ser....
    • Dong, H.J., Liu, Y.Z.: Weighted mixed norm estimates for fractional wave equations with VMO coefficients. J. Differ. Equ. 337, 168–254 (2022) Article...
    • Dong, H.J., Liu, Y.Z.: Sobolev estimates for fractional parabolic equations with space-time non-local operators. Calc. Var. Part. Differ....
    • Farkas, W., Jacob, N., Schilling, R.L.: Function spaces related to continuous negative definite functions: -Bessel potential spaces. Dissertationes...
    • Fogedby, H.C.: Lévy flights in random environments. Phys. Rev. Lett. 73, 2517–2520 (1994) Article Google Scholar
    • Fujita, Y.: Integrodifferential equation which interpolates the heat equation and the wave equation. Osaka J. Math. 27, 309–321 (1990) MathSciNet...
    • Grafakos, L.: Classical Fourier Analysis. Springer, Berlin (2008) Book Google Scholar
    • Guan, Q.-Y.: Integration by parts formula for regional fractional Laplacian. Commun. Math. Phys. 266, 289–329 (2006) Article MathSciNet Google...
    • Hanyga, A.: Multidimensional solutions of space-fractional diffusion equations. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 457, 2993–3005...
    • He, J.W., Zhou, Y.: On a backward problem for nonlinear time fractional wave equations. Proc. Roy. Soc. Edinb. Sect. A 152, 1589–1612 (2022) Article...
    • He, J.W., Zhou, Y.: Local/global existence analysis of fractional wave equations with exponential nonlinearity. Bull. Sci. Math. 189, 103357...
    • Hilfer, R.: Applications of Fractional Calculus in Physics. World Scientific, Singapore (2000) Book Google Scholar
    • Hirata, H., Miao, C.X.: Space-time estimates of linear flow and application to some nonlinear integro-differential equations corresponding...
    • Hörmander, L.: Estimates for translation invariant operators in spaces. Acta. Math. 104, 93–140 (1960) Article MathSciNet Google Scholar
    • Hörmander, L.: The Analysis of Linear Partial Differential Operators I. Springer, Berlin (1983) Google Scholar
    • Kang, J., Park, D.: An -theory for space-time non-local equations generated by Lévy processes with low intensity of small jumps. Stoch. PDE...
    • Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies,...
    • Kim, I., Kim, K.-H., Kim, P.: Parabolic Littlewood–Paley inequality for -type operators and applications to stochastic integro-differential...
    • Kim, K.-H., Park, D., Ryu, J.: An -theory for diffusion equations with space-time nonlocal operators. J. Differ. Equ. 287, 376–427 (2021) Article...
    • Kim, P., Song, R.M., Vondracek, Z.: Global uniform boundary Harnack principle with explicit decay rate and its application. Stoch. Proc. Appl....
    • Kim, P., Song, R.M., Vondracek, Z.: Potential theory of subordinate killed Brownian motion. Trans. Am. Math. Soc. 371, 3917–3969 (2019) Article...
    • Li, Y.J., Wang, Y.J., Deng, W.H.: Galerkin finite element approximations for stochastic space-time fractional wave equations. SIAM J. Numer....
    • Louis-Rose, C., Warma, M.: Approximate controllability from the exterior of space-time fractional wave equations. Appl. Math. Optim. 83, 207–250...
    • Mainardi, F.: Fractional Calculus and Waves in Linear Viscoelasticity. An Introduction to Mathematical Models, Imperial College Press, London...
    • Metzler, R., Klafter, J.: The random walk’s guide to anomalous diffusion: a fractional dynamics approach. Phys. Rep. 339, 1–77 (2000) Article...
    • Mikulevicius, R., Phonsom, C.: On the Cauchy problem for integro-differential equations in the scale of spaces of generalized smoothness....
    • Otárola, E., Salgado, A.J.: Regularity of solutions to space-time fractional wave equations: a PDE approach. Fract. Calc. Appl. Anal. 21,...
    • Ribaud, F.: Cauchy problem for semilinear parabolic equations with initial data in spaces. Rev. Mat. Iberoamericana 14, 1–46 (1998) Article...
    • Sandev, T., Tomovski, Z.: Fractional Equations and Models: Theory and Applications. Springer, Berlin (2019) Book Google Scholar
    • Sato, K.-I.: Lévy Processes and Infinitely Divisible Distributions. Cambridge University Press, Cambridge (1999) Google Scholar
    • Schilling, R.L., Song, R.M., Vondracek, Z.: Bernstein Functions: Theory and Applications. De Gruyter Studies in Mathematics, vol. 37, 2nd...
    • Zaslavsky, G.M., Edelman, M., Niyazov, B.A.: Self-similarity, renormalization, and phase space nonuniformity of Hamiltonian chaotic dynamics....
    • Zhang, Q., Li, Y.: Global well-posedness and blow-up solutions of the Cauchy problem for a time-fractional superdiffusion equation. J. Evol....
    • Zhou, Y.: Basic Theory of Fractional Differential Equations. World Scientific, Singapore (2014) Book Google Scholar
    • Zhou, Y.: Fractional Evolution Equations and Inclusions: Analysis and Control. Academic Press, San Diego (2016)
    • Zhou, Y., He, J.W.: Well-posedness and regularity for fractional damped wave equations. Monatsh. Math. 194, 425–458 (2021) Article MathSciNet...
    • Zhou, Y., He, J.W., Alsaedi, A., Ahmad, B.: The well-posedness for semilinear time fractional wave equations on . Elec. Res. Arch. 30, 2981–3003...
Fundación Dialnet

Mi Documat

Opciones de artículo

Opciones de compartir

Opciones de entorno