Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Finite time blow-up for the inhomogeneous equation $u_{t}=\Delta u+a(x)u^{p}+\lambda \phi $ in $R^{d}$


Author: Ross G. Pinsky
Journal: Proc. Amer. Math. Soc. 127 (1999), 3319-3327
MSC (1991): Primary 35K15, 35K55
DOI: https://doi.org/10.1090/S0002-9939-99-05164-3
Published electronically: May 17, 1999
MathSciNet review: 1641081
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We consider the inhomogeneous equation

\begin{equation*}\begin{split} & u_{t}=\Delta u+a(x)u^{p}+\lambda \phi (x) \ \text{in} \ R^{d}, t\in (0,T),\\ &u(x,0)=f(x),\end{split}\end{equation*}

where $a,\phi \gneqq 0$, $\lambda >0$ and $f\ge 0$, and give criteria on $p,d,a$, and $\phi $ which determine whether for all $\lambda $ and all $f$ the solution blows up in finite time or whether for $\lambda $ and $f$ sufficiently small, the solution exists for all time.


References [Enhancements On Off] (What's this?)

  • 1. Gilbarg, D. and Trudinger, N. S., Elliptic Partial Differential Equations of Second Order, Springer-Verlag, Berlin/New York, 1983. MR 86c:35035
  • 2. Lee T.Y., Some limit theorems for super-Brownian motion and semilinear differential equations, Annals of Probab. 21 (1993), 979-995. MR 94b:60038
  • 3. Levine H., The role of critical exponents in blowup theorems, SIAM Rev 32 (1990), 262-288. MR 91j:35135
  • 4. Pazy A., Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, Berlin/New York, 1983. MR 85g:47061
  • 5. Pinsky R., Existence and Nonexistence of Global Solutions for $u_{t}=\Delta u+a(x)u^{p}$ in $R^{d}$, J.Diff Equas. 133 (1997), 152-177. MR 97k:35118
  • 6. Zhang Q. S., A new critical phenomenon for semilinear parabolic problems, Jour. Math. Anal. and App. 219 (1998), 125-139. MR 98m:35095

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 35K15, 35K55

Retrieve articles in all journals with MSC (1991): 35K15, 35K55


Additional Information

Ross G. Pinsky
Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000, Israel
Email: pinsky@tx.technion.ac.il

DOI: https://doi.org/10.1090/S0002-9939-99-05164-3
Keywords: Finite time blow-up, semilinear reaction-diffusion equations, critical exponent
Received by editor(s): February 11, 1998
Published electronically: May 17, 1999
Additional Notes: This research was supported by the Fund for the Promotion of Research at the Technion.
Communicated by: Lesley M. Sibner
Article copyright: © Copyright 1999 American Mathematical Society

American Mathematical Society