Finite time blow-up and global solutions for semilinear parabolic equations with initial data at high energy level

By Filippo Gazzola, Tobias Weth. Differential and Integral Equations, 2005

146 citations recorded by the source corpus.

For a class of semilinear parabolic equations on a bounded domain Ω, we analyze the behavior of the solutions when the initial data varies in the phase space H 1(Ω). We obtain both global solutions and finite time blow-up solutions. Our main tools are the comparison principle and variational methods. Particular attention is paid to initial data at high energy level; to this end, a basic new idea is to exploit the weak dissipativity (respectively antidissipativity) of the semiflow inside (respectively outside) the Nehari manifold.

Subfields: Applied Mathematics, Control and Systems Engineering, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/de4297e650b3850e9d81bd1e35b04ca990dd26cf

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Jun Cao
2. Binlin Zhang
3. Marco Squassina
4. Ning Pan
5. Carlos A. Escudero
6. I. Peral
7. Hua Chen
8. Gongwei Liu

In Control and Systems Engineering

1. Jun Cao
2. Ning Pan
3. Marco Squassina
4. Binlin Zhang
5. Hua Chen
6. Filippo Gazzola
7. I. Peral
8. Runzhang Xu

In Mathematical Physics

1. Binlin Zhang
2. Ning Pan
3. Filippo Gazzola
4. Jun Cao
5. Hua Chen
6. Marco Squassina
7. I. Peral
8. Runzhang Xu

Citations in the corpus

• Global solutions and finite time blow up for damped semilinear wave equations ? ? The first author w (2006, background): “In such cas e.g. [24, Section 3]), any mountain pass solutionu of the stationary problem is a minimizer for (2.2) (i.e. it satisfi ‖∇u‖22 = Sp‖u‖2p) andSp is related to its energy d = p − 2 2p S p/(p−2) p .”
• Global existence versus blow-up results for a fourth order parabolic PDE involving the Hessian (2015, background): “What happens for J(u0) > d? Possible hints may be found in [16, 19] although the lack of a comparison principle for (9) certainly creates more difficulties.”
• Global existence and nonexistence for semilinear parabolic equations with conical degeneration (2012, background): “Recently, there are some interesting results about the global existence and blow-up in [5] in which the authors proved for low energy data u0, i.”
• Degenerate Kirchhoff-type diffusion problems involving the fractional p-Laplacian (2017, background): “Gazzola and Weth [12] studied the initial-boundary value problem of (1.5), where f(u) = |u|p−1u.”