高阶抛物不等式组的Liouville型定理及其应用

In this paper, we establish a Liouville-type theorem for a system of higher-order parabolic inequalities by using the method of test functions and an integral estimate. As an application, we observe the Fujita blow-up phenomena for the corresponding parabolic system, which in particular fills up the gap in the recent result of Pang et. al. （Existence and nonexistence of global solutions for a higher-order semilinear parabolic system, Indiana Univ. Math. J., 55（2006）, 1113-1134）. Moreover, the importance of this observation is that we do not impose any regularity assumption on the initial data.     

Asymptotic Behavior of Solutions to a Class of Semilinear Parabolic Equations with Boundary Degeneracy

This paper concerns the asymptotic behavior of solutions to one-dimensional semilinear parabolic equations with boundary degeneracy both in bounded and unbounded intervals. For the problem in a bounded interval, it is shown that there exist both nontrivial global solutions for small initial data and blowing-up solutions for large one if the degeneracy is not strong. Whereas in the case that the degeneracy is strong enough, the nontrivial solution must blow up in a finite time. For the problem in an unbounded interval, blowing-up theorems of Fujita type are established. It is shown that the critical Fujita exponent depends on the degeneracy of the equation and the asymptotic behavior of the diffusion coefficient at infinity, and it may be equal to one or infinity. Furthermore, the critical case is proved to belong to the blowing-up case.     

Cauchy problems of semilinear pseudo-parabolic equations

This paper concerns with the Cauchy problems of semilinear pseudo-parabolic equations. After establishing the necessary existence, uniqueness and comparison principle for mild solutions, which are also classical ones provided that the initial data are appropriately smooth, we investigate large time behavior of solutions. It is shown that there still exist the critical global existence exponent and the critical Fujita exponent for pseudo-parabolic equations and that these two critical exponents are consistent with the corresponding semilinear heat equations.     

A note on semilinear heat equation in hyperbolic space

Bandle et al. [1] obtained a quite interesting result about a semilinear heat equation that the Fujita exponent relative to the whole hyperbolic space is just the same as that relative to bounded domain in Euclidean space, and, in addition, the properties of solutions are different in the critical exponent case. Our purpose is to answer an open problem proposed by Bandle et al. for the critical exponent case, and it, together with the one obtained by them, shows that the critical exponent case does belong to the non-blow-up case, which is completely different from the case in Euclidean space.     

Cauchy problem for a class of fourth-order semilinear pseudohyperbolic equations with structural damping

A sufficient condition for the existence of global solutions is established, and the rate of decay of solutions to the Cauchy problem for a semilinear pseudohyperbolic equation with structural damping is found. The nonexistence of global solutions is also investigated. An analogue of the critical Fujita exponent is obtained for the considered problem.     

含奇异项的半线性抛物方程组的Cauchy问题

本文考察含奇异项的半线性抛物方程组的Ｃａｕｃｈｙ问题,算出了该问题的猝灭临界指标和猝灭临界维数．     

Blow-up rates for higher-order semilinear parabolic equations and systems and some Fujita-type theorems

In this paper, we derive blow-up rates for higher-order semilinear parabolic equations and systems. Our proof is by contradiction and uses a scaling argument. This procedure reduces the problems of blow-up rate to Fujita-type theorems. In addition, we also give some new Fujita-type theorems for higher-order semilinear parabolic equations and systems with the time variable on R. These results are not restricted to positive solutions.     

Fujita phenomenon in inhomogeneous fast diffusion system

This paper deals with the Fujita phenomenon for the Cauchy problem of an inhomogeneous fast diffusion system. Both the critical exponent and the second exponent are obtained. We observe that the inhomogeneous terms in the system substantially contribute to the critical exponent, in that the blow-up exponent region is obviously enlarged, with keeping the second critical exponent unchanged for small inhomogeneous sources.     

Global existence of solutions for a p-Laplacian equation with nonlocal Fisher–KPP type reaction terms

This work is concerned with the Neumann initial boundary value problem and Cauchy problem of a parabolic p-Laplacian equation with nonlocal Fisher–KPP type reaction terms. We establish a uniform boundedness and global existence of solutions to the equation by applying the method of multipliers and modified Moser's iteration technique for some ranges of parameters. The ranges of parameters have similar structure to that of the classical critical Fujita exponent.     

一类双重退化抛物方程组解的整体存在与爆破

研究一类具有非线性边界流的双重退化抛物方程组解的整体存在与爆破,通过构造自相似的上下解,得到了整体存在曲线.借助一些新的结果,获得了Fujita临界指数.其中一个有趣的现象是:整体存在曲线和Fujita临界曲线分别是由一个矩阵和线性方程组来决定.     

On the asymptotic behavior of unbounded radial solutions for semilinear parabolic problems involving critical Sobolev exponent

This paper is concerned with a semilinear parabolic equation involving critical Sobolev exponent in a ball or in RN. The asymptotic behavior of unbounded, radially symmetric, nonnegative global solutions which do not decay to zero is given. The structure of the space of initial data is also discussed.     

高阶半线性抛物型方程组的生命跨度

其中m,P,q>1．利用试验函数方法,首先推导一些积分不等式,然后对方程组爆破解的生命跨度 [0,T)给出估计．     

Fujita exponents for a space–time weighted parabolic problem in bounded domains

This paper deals with a semilinear weighted parabolic problem in general bounded domains, subject to zero Dirichlet boundary conditions, where the weighted functions depend not only on space variable but also on time variable. Fujita exponents for blow-up and global existence of solutions are determined by using semigroup methods and the comparison principle, which are composed by the dimensions of the space domains and the eight exponents in nonlinear coupled sources and the space–time weighted functions.     

Critical exponents of the non-Newtonian polytropic filtration equation with nonlinear boundary condition

This work is concerned with the critical exponent of the non-Newtonian polytropic filtration equation with nonlinear boundary conditions. We obtain the critical global existence exponent and critical Fujita exponent by constructing various self-similar supersolutions and subsolutions.     

Algebraic criteria for global existence or blow-up for a boundary coupled system of nonlinear diffusion equations

This article deals with the critical curves for a degenerate parabolic system coupled via nonlinear boundary flux. By constructing the self-similar supersolution and subsolution, we obtain the critical global existence curve. The critical curve of Fujita type is conjectured with the aid of some new results.     

Large time behavior of solutions to Newtonian filtration equation with nonlinear boundary sources

This paper deals with the exterior problem of the Newtonian filtration equation with nonlinear boundary sources. The large time behavior of solutions including the critical Fujita exponent are determined or estimated. An interesting phenomenon is illustrated that there exists a threshold value for the coefficient of the lower order term, which depends on the spacial dimension. Exactly speaking, the critical global exponent is strictly less than the critical Fujita exponent when the coefficient is under this threshold, while these two exponents are identically equal when the coefficient is over this threshold. Supported by the NNSF of China and the China Postdoctoral Science Foundation.     

Critical exponents in a degenerate parabolic equation with weighted source

This article deals with the Fujita-type theorems to the Cauchy problem of degenerate parabolic equation not in divergence form with weighted source u _t ?=?u ^p Δu?+?a(x)u ^q in ? ⁿ ?×?(0,?T), where p?≥?1, q?>?1, and the positive weight function a(x) is of the order |x| ^m with m?>??2. It was known that for the degenerate diffusion equation in divergence form, the weight function affects both of the critical Fujita exponent and the second critical exponent (describing the critical smallness of initial data required by global solutions via the decay rates of the initial data at space-infinity). Contrarily, it is interesting to prove that the weight function in the present model with degenerate diffusion not in divergence form influences the second critical exponent only, without changing the critical Fujita exponent.     

Fujita type exponent for fully nonlinear parabolic equations and existence results

In this paper, we prove that a class of parabolic equations involving a second order fully nonlinear uniformly elliptic operator has a Fujita type exponent. These exponents are related with an eigenvalue problem in all RN and play the same role in blow-up theorems as the classical p^?=1+2/N introduced by Fujita for the Laplacian. We also obtain some associated existence results.     

Homogenization of Semilinear Parabolic Operators in a Perforated Cylinder

In this paper, we consider a semilinear parabolic equation of second order with lower term a power function of the unknown function and prove that the sequence of solutions in a perforated cylinder tends to a solution in a unperforated cylinder if the radii of rejected balls in the parabolic metric tend to zero at the rate depending on the exponent of the power function in the lower term.     

Solutions with moving singularities for a semilinear parabolic equation

We consider the Cauchy problem for a semilinear heat equation with power nonlinearity. It is known that the equation has a singular steady state in some parameter range. Our concern is a solution with a moving singularity that is obtained by perturbing the singular steady state. By formal expansion, it turns out that the remainder term must satisfy a certain parabolic equation with inverse-square potential. From the well-posedness of this equation, we see that there appears a critical exponent. Paying attention to this exponent, for a prescribed motion of the singular point and suitable initial data, we establish the time-local existence, uniqueness and comparison principle for such singular solutions. We also consider solutions with multiple singularities.