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Asymptotic stability of wave equations with memory and frictional boundary dampings

Fatiha Alabau-Boussouira (2008)

Applicationes Mathematicae

This work is concerned with stabilization of a wave equation by a linear boundary term combining frictional and memory damping on part of the boundary. We prove that the energy decays to zero exponentially if the kernel decays exponentially at infinity. We consider a slightly different boundary condition than the one used by M. Aassila et al. [Calc. Var. 15, 2002]. This allows us to avoid the assumption that the part of the boundary where the feedback is active is strictly star-shaped. The result...

Asymptotics and stability for global solutions to the Navier-Stokes equations

Isabelle Gallagher, Dragos Iftimie, Fabrice Planchon (2003)

Annales de l’institut Fourier

We consider an a priori global strong solution to the Navier-Stokes equations. We prove it behaves like a small solution for large time. Combining this asymptotics with uniqueness and averaging in time properties, we obtain the stability of such a global solution.

Blow-up of weak solutions for the semilinear wave equations with nonlinear boundary and interior sources and damping

Lorena Bociu, Irena Lasiecka (2008)

Applicationes Mathematicae

We focus on the blow-up in finite time of weak solutions to the wave equation with interior and boundary nonlinear sources and dissipations. Our central interest is the relationship of the sources and damping terms to the behavior of solutions. We prove that under specific conditions relating the sources and the dissipations (namely p > m and k > m), weak solutions blow up in finite time.

Blowup rates for nonlinear heat equations with gradient terms and for parabolic inequalities

Philippe Souplet, Slim Tayachi (2001)

Colloquium Mathematicae

Consider the nonlinear heat equation (E): u t - Δ u = | u | p - 1 u + b | ∇ u | q . We prove that for a large class of radial, positive, nonglobal solutions of (E), one has the blowup estimates C ₁ ( T - t ) - 1 / ( p - 1 ) ≤ | | u ( t ) | | ∞ ≤ C ₂ ( T - t ) - 1 / ( p - 1 ) . Also, as an application of our method, we obtain the same upper estimate if u only satisfies the nonlinear parabolic inequality u t - u x x ≥ u p . More general inequalities of the form u t - u x x ≥ f ( u ) with, for instance, f ( u ) = ( 1 + u ) l o g p ( 1 + u ) are also treated. Our results show that for solutions of the parabolic inequality, one has essentially the same estimates as for solutions of the ordinary...

Blowup solutions to Keller-Segel system and its simplified systems

Takasi Senba (2006)

Banach Center Publications

In this paper, we will consider blowup solutions to the so called Keller-Segel system and its simplified form. The Keller-Segel system was introduced to describe how cellular slime molds aggregate, owing to the motion of the cells toward a higher concentration of a chemical substance produced by themselves. We will describe a common conjecture in connection with blowup solutions to the Keller-Segel system, and some results for solutions to simplified versions of the Keller-Segel system, giving the...

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