Critical Fujita Exponents for Localized Reaction Diffusion Systems

In this paper, we prove the existence of cirtical Fujita exponents for a class of localized reaction diffusion systems.

Critical Exponents for Fast Diffusion Equations with Nonlinear Boundary Sources

In this paper,we study the large time behavior of solutions to a class of fast diffusion equations with nonlinear boundary sources on the exterior domain of the unit ball.We are interested in the critical global exponent q_o and the critical Fujita exponent q_c for the problem considered,and show that q_o=q_c for the multidimensional Non-Newtonian polytropic filtration equation with nonlinear boundary sources,which is quite different from the known results that q_o〈q_c for the onedimensional case;moreover,the value is different from the slow case.

Fujita phenomena in nonlinear pseudo-parabolic system

This paper deals with the Cauchy problem to the nonlinear pseudo-parabolic system ut - △u - αut =vp, vt -△v - α△vt = uq with p, q≥ 1 and pq 〉 1, where the viscous terms of third order are included. We first find the critical Fujita exponent, and then determine the second critical exponent to characterize the critical space-decay rate of initial data in the co-existence region of global and non-global solutions. Moreover, time-decay profiles are obtained for the global solutions. It can be found that, different from those for the situations of general semilinear heat systems, we have to use distinctive techniques to treat the influence from the viscous terms of the highest order. To fix the non-global solutions, we exploit the test function method, instead of the general Kaplan method for heat systems. To obtain the global solutions, we apply the LP-Lq technique to establish some uniform Lm time-decay estimates. In particular, under a suitable classification for the nonlinear parameters and the initial data, various Lm time-decay estimates in the procedure enable us to arrive at the time-decay profiles of solutions to the system. It is mentioned that the general scaling method for parabolic problems relies heavily on regularizing effect to establish the compactness of approximating solutions, which cannot be directly realized here due to absence of the smooth effect in the pseudo-parabolic system.

Fujita-Liouville Type Theorem for Coupled Fourth-Order Parabolic Inequalities

This paper deals with a coupled system of fourth-order parabolic inequalities ｜u｜t ≥ -△2^u＋｜v｜^q, ｜v｜t ≥-△2v＋｜u｜p^ in S=R^n ×R^＋ withp, q 〉 1, n ≥1. AFujita- Liouville type theorem is established that the inequality system does not admit nontrivial nonnegative global solutions on S whenever n/4≤ max（ p＋1/pq-1, q＋1/pq-1 ）. Since the general maximum-comparison principle does not hold for the fourth-order problem, the authors use the test function method to get the global non-existence of nontrivial solutions.

Global and Nonglobal Solutions for Pseudo-Parabolic Equation with Inhomogeneous Terms

This paper considers the Cauchy problem of pseudo-parabolic equation with inhomogeneous terms u_(t)=△u+k△u_(t)+w(x)u^(P)(x,t).In[1],Li et al.gave the critical Fujita exponent,second critical exponent and the life span for blow-up solutions under w(x)=|x|^(σ)with>0.We further generalize the weight function w(x)~|x|^(σ)for-2<σ<0,and discuss the global and non-global solutions to obtain the critical Fujita exponent.