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Evolution PDEs with Nonstandard Growth Conditions

Existence, Uniqueness, Localization, Blow-up

  • Book
  • © 2015

Overview

Authors:
  1. Stanislav Antontsev

    1. Center for Mathematics & Fundamental Applications, University of Lisbon, Lisbon, Portugal

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    You can also search for this author in PubMed   Google Scholar

  2. Sergey Shmarev

    1. Mathematics, University of Oviedo, Oviedo, Spain

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    You can also search for this author in PubMed   Google Scholar

  • The first systematic exposition of the theory of evolution PDEs with nonstandard growth conditions
  • Specific tools for the study of PDEs with nonstandard growth are developed
  • New localization effects due to anisotropy and variable nonlinearity are discovered
  • A number of new not previously published results is included
  • A review of other approaches and known results is provided
  • Includes supplementary material: sn.pub/extras

Part of the book series: Atlantis Studies in Differential Equations (ASDE, volume 4)

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Table of contents (12 chapters)

  1. Front Matter

    Pages i-xvii
    Download chapter PDF

  2. The Function Spaces

    • Stanislav Antontsev, Sergey Shmarev
    Pages 1-43

  3. A Porous Medium Equation with Variable Nonlinearity

    • Stanislav Antontsev, Sergey Shmarev
    Pages 45-80

  4. Localization of Solutions of the Generalized Porous Medium Equation

    • Stanislav Antontsev, Sergey Shmarev
    Pages 81-105

  5. Anisotropic Equations with Variable Growth and Coercivity Conditions

    • Stanislav Antontsev, Sergey Shmarev
    Pages 107-184

  6. Space Localization of Energy Solutions

    • Stanislav Antontsev, Sergey Shmarev
    Pages 185-217

  7. Extinction in a Finite Time and the Large Time Behavior

    • Stanislav Antontsev, Sergey Shmarev
    Pages 219-240

  8. Blow-up in Equations with Variable Nonlinearity

    • Stanislav Antontsev, Sergey Shmarev
    Pages 241-265

  9. Equations with Double Isotropic Nonlinearity

    • Stanislav Antontsev, Sergey Shmarev
    Pages 267-289

  10. Strong Solutions of Doubly Nonlinear Anisotropic Equations

    • Stanislav Antontsev, Sergey Shmarev
    Pages 291-318

  11. Anisotropic Equations with Double Nonlinearity: Blow-up and Vanishing

    • Stanislav Antontsev, Sergey Shmarev
    Pages 319-341

  12. Wave Equation with \(p(x,t)\)-Laplacian

    • Stanislav Antontsev, Sergey Shmarev
    Pages 343-368

  13. Semilinear Hyperbolic Equations

    • Stanislav Antontsev, Sergey Shmarev
    Pages 369-391

  14. Back Matter

    Pages 393-409
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Keywords

About this book

This monograph offers the reader a treatment of the theory of evolution PDEs with nonstandard growth conditions. This class includes parabolic and hyperbolic equations with variable or anisotropic nonlinear structure. We develop methods for the study of such equations and present a detailed account of recent results. An overview of other approaches to the study of PDEs of this kind is provided. The presentation is focused on the issues of existence and uniqueness of solutions in appropriate function spaces and on the study of the specific qualitative properties of solutions, such as localization in space and time, extinction in a finite time and blow-up, or nonexistence of global in time solutions. Special attention is paid to the study of the properties intrinsic to solutions of equations with nonstandard growth.

Authors and Affiliations

  • Center for Mathematics & Fundamental Applications, University of Lisbon, Lisbon, Portugal

    Stanislav Antontsev

  • Mathematics, University of Oviedo, Oviedo, Spain

    Sergey Shmarev

Bibliographic Information

  • Book Title: Evolution PDEs with Nonstandard Growth Conditions

  • Book Subtitle: Existence, Uniqueness, Localization, Blow-up

  • Authors: Stanislav Antontsev, Sergey Shmarev

  • Series Title: Atlantis Studies in Differential Equations

  • DOI: https://doi.org/10.2991/978-94-6239-112-3

  • Publisher: Atlantis Press Paris

  • eBook Packages: Mathematics and Statistics, Mathematics and Statistics (R0)

  • Copyright Information: The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature B.V. 2015

  • Hardcover ISBN: 978-94-6239-111-6Published: 14 April 2015

  • eBook ISBN: 978-94-6239-112-3Published: 01 April 2015

  • Series ISSN: 2214-6253

  • Series E-ISSN: 2214-6261

  • Edition Number: 1

  • Number of Pages: XVII, 409

  • Number of Illustrations: 1 b/w illustrations

  • Topics: Partial Differential Equations, Functional Analysis

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