Finite time blow-up for the inhomogeneous equation $u_{t}=\Delta u+a(x)u^{p}+\lambda \phi$ in $R^{d}$
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- by Ross G. Pinsky PDF
- Proc. Amer. Math. Soc. 127 (1999), 3319-3327 Request permission
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
- David Gilbarg and Neil S. Trudinger, Elliptic partial differential equations of second order, 2nd ed., Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 224, Springer-Verlag, Berlin, 1983. MR 737190, DOI 10.1007/978-3-642-61798-0
- Tzong-Yow Lee, Some limit theorems for super-Brownian motion and semilinear differential equations, Ann. Probab. 21 (1993), no. 2, 979–995. MR 1217576
- Howard A. Levine, The role of critical exponents in blowup theorems, SIAM Rev. 32 (1990), no. 2, 262–288. MR 1056055, DOI 10.1137/1032046
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
- Ross G. Pinsky, Existence and nonexistence of global solutions for $u_t=\Delta u+a(x)u^p$ in $\textbf {R}^d$, J. Differential Equations 133 (1997), no. 1, 152–177. MR 1426761, DOI 10.1006/jdeq.1996.3196
- Qi S. Zhang, A new critical phenomenon for semilinear parabolic problems, J. Math. Anal. Appl. 219 (1998), no. 1, 125–139. MR 1607106, DOI 10.1006/jmaa.1997.5825
Additional Information
- Ross G. Pinsky
- Affiliation: Department of Mathematics, Technion-Israel Institute of Technology, Haifa, 32000, Israel
- Email: pinsky@tx.technion.ac.il
- 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
- © Copyright 1999 American Mathematical Society
- 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
- MathSciNet review: 1641081