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Blow up, global existence and growth rate estimates in nonlinear parabolic systems

Joanna Rencławowicz (2000)

Colloquium Mathematicae

We prove Fujita-type global existence and nonexistence theorems for a system of m equations (m > 1) with different diffusion coefficients, i.e. u i t - d i Δ u i = k = 1 m u k p k i , i = 1 , . . . , m , x N , t > 0 , with nonnegative, bounded, continuous initial values and p k i 0 , i , k = 1 , . . . , m , d i > 0 , i = 1 , . . . , m . For solutions which blow up at t = T < , we derive the following bounds on the blow up rate: u i ( x , t ) C ( T - t ) - α i with C > 0 and α i defined in terms of p k i .

Cauchy problem for a class of parabolic systems of Shilov type with variable coefficients

Vladyslav Litovchenko, Iryna Dovzhytska (2012)

Open Mathematics

In the case of initial data belonging to a wide class of functions including distributions of Gelfand-Shilov type we establish the correct solvability of the Cauchy problem for a new class of Shilov parabolic systems of equations with partial derivatives with bounded smooth variable lower coefficients and nonnegative genus. We also investigate the conditions of local improvement of the convergence of a solution of this problem to its limiting value when the time variable tends to zero.

Coherent nonlinear waves and the Wiener algebra

Guy Métivier, Jean-Luc Joly, Jeffrey Rauch (1994)

Annales de l'institut Fourier

We study oscillatory solutions of semilinear first order symmetric hyperbolic system L u = f ( t , x , u , u ) , with real analytic f .The main advance in this paper is that it treats multidimensional problems with profiles that are almost periodic in T , X with only the natural hypothesis of coherence.In the special case where L has constant coefficients and the phases are linear, the solutions have asymptotic description u ϵ = U ( t , x , t / ϵ , x / ϵ ) + o ( 1 ) where the profile U ( t , x , T , X ) is almost periodic in ( T , X ) .The main novelty in the analysis is the space of profiles which...

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