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Mediterranean Journal of Mathematics

, Volume 10, Issue 1, pp 289–311 | Cite as

Strong Regularizing Effect of a Gradient Term in the Heat Equation with a Weight

  • Boumediene Abdellaoui
  • Yasmina Nasri
  • Ana PrimoEmail author
Article
  • 122 Downloads

Abstract

We deal with the following parabolic problem,
$$(P)\left\{\begin{array}{lll} u_t - \Delta{u} + |\nabla{u}|^q \quad=\quad \lambda{g}(x)u + f(x, t),\quad u > 0 \; {\rm in} \; \Omega \; \times \; (0, T),\\ \qquad\quad\quad\; u(x, t) \quad=\quad 0 \quad{\rm on}\; {\partial}{\Omega}\; \times ; (0, T),\\ \qquad\quad\quad\; u(x, 0) \quad=\quad u_{0}(x), \quad x \in {\Omega},\end{array}\right.$$
where Open image in new window is a bounded regular domain or \({\Omega = \mathbb{R}^N}\) , \({1 < q \leq 2, \lambda > 0\; {\rm and}\; f \geq 0, u_{0} \geq 0}\) are in a suitable class of functions. We give assumptions on g with respect to q for which for all λ >  0 and all \({f \in L^1(\Omega_T ), f \geq 0}\) , problem (P) has a positive solution.

Under some additional conditions on the data, the Cauchy problem and the asymptotic behavior of the solution are also considered.

Mathematics Subject Classification (2010)

35D05 35D10 35J20 35J25 35J70 46E30 46E35 

Keywords

Semilinear heat equations with gradient term regularization comparison arguments 

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Copyright information

© Springer Basel AG 2012

Authors and Affiliations

  • Boumediene Abdellaoui
    • 1
  • Yasmina Nasri
    • 1
  • Ana Primo
    • 2
    Email author
  1. 1.Dèpartement de MathèmatiquesUniversitè Aboubekr Belkaïd, TlemcenTlemcenAlgeria
  2. 2.ICMAT (Instituto de Ciencias Matemáticas)MadridSpain

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