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Exponential decay and blow-up results for a nonlinear heat equation with a viscoelastic term and Robin conditions

Volume 119 / 2017

Nguyen Thanh Long, Nguyen Van Y, Le Thi Phuong Ngoc Annales Polonici Mathematici 119 (2017), 121-145 MSC: 34B60, 35K55, 35Q72, 80A30. DOI: 10.4064/ap4084-3-2017 Published online: 14 June 2017

Abstract

We consider a nonlinear heat equation with a viscoelastic term and Robin conditions. First, we prove existence and uniqueness of a weak solution. Next, we prove that any weak solution with negative initial energy will blow up in finite time. Finally, we give a sufficient condition for the global existence and exponential decay of weak solutions. The main tools are the Faedo–Galerkin method and defining a modified energy functional together with the technique of Lyapunov functional.

Authors

  • Nguyen Thanh LongDepartment of Mathematics and Computer Science
    University of Natural Science
    Vietnam National University Ho Chi Minh City
    227 Nguyen Van Cu St., Dist. 5
    Ho Chi Minh City, Vietnam
    e-mail
  • Nguyen Van YDepartment of Fundamental Sciences
    Ho Chi Minh City University of Food Industry
    140 Le Trong Tan Str., Tay Thanh Ward, Tan Phu Dist.
    Ho Chi Minh City, Vietnam
    and
    Department of Mathematics and Computer Science
    University of Natural Science
    Vietnam National University Ho Chi Minh City
    227 Nguyen Van Cu St., Dist. 5
    Ho Chi Minh City, Vietnam
    e-mail
  • Le Thi Phuong NgocUniversity of Khanh Hoa
    01 Nguyen Chanh St.
    Nha Trang City, Vietnam
    e-mail

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