This paper deals with the higher dimension quasilinear parabolic-parabolic Keller-Segel system involving a source term of logistic typein Ω × (0, T ), subject to nonnegative initial data and homogeneous Neumann boundary condition, where Ω is smooth and bounded domain in R n , n ≥ 2, φ and g are smooth and positive functions satisfyingIt was known that the model without the logistic source admits both bounded and unbounded solutions, identified via the critical exponent 2 n . On the other hand, the model is just a critical case with the balance of logistic damping and aggregation effects, for which the property of solutions should be determined by the coefficients involved. In the present paper it is proved that there is θ 0 > 0 such that the problem admits global bounded classical solutions, regardless of the size of initial data and diffusion whenever χ µ < θ 0 . This shows the substantial effect of the logistic source to the behavior of solutions.
We establish the critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux and then determine the blow-up rates and the blow-up sets for the nonglobal solutions. 2004 Elsevier Inc. All rights reserved.
We establish the blow-up rate for the solution of a nonlinear diffusion equation (u m)t = uxx, 0 < x < 1, t > 0, subject to Neumann boundary conditions ux(0, t) = 0, ux(1, t) = u α (1, t).
This paper deals with a nonlinear diffusion system coupled via nonlinear reaction terms of power type. As results of interactions among the multi-nonlinearities in the system described by six exponents, global boundedness and blow-up criteria of positive solutions are determined.
Mathematics Subject Classification (2000). Primary 35K65, 35K40.
This paper deals with a doubly degenerate parabolic system multi-coupled by inner and boundary sources. The necessary-sufficient conditions for global weak solutions are determined, which involve a complete classification for all the eight nonlinear parameters of the model and cover all possible blowing up mechanisms of solutions. The results of the paper are mainly rely on the comparison principle and the energy method. (2000). Primary 35B33 · 35K57 · 35K65.
Mathematics Subject Classification
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