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This paper is concerned with the time delayed Nicholson’s blowflies equation with degenerate diffusion. We prove the existence and uniqueness of the positive steady state solution under the Dirichlet boundary condition and we show the stability of the nontrivial steady state.

This paper is concerned with the bifurcation phenomenon of a free-boundary problem modeling the tumor growth under the action of angiogenesis and inhibitor. Taking the surface tension coefficient γ as a bifurcation parameter, we prove that there exist a positive integer $m^{**}$ and a sequence of $\gamma_{m}$ such that, for every $\gamma_{m}$ ( $m>m^{**}$ ), symmetry-breaking stationary solutions bifurcate from the radially symmetric stationary solutions.

This paper is concerned with a class of nonlinear diffusion equations coupled by the nonlinear boundary sources on the exterior domain of the unit ball in \(\mathbb R^N\). We are interested in the critical curves which can describe the large time behavior of the solutions. It is shown that the critical global existence curve coincides with the critical Fujita curve for the system we considered.

This paper deals with a free boundary problem modeling tumor growth with inhibitors. This problem has a unique radially symmetric stationary solution with radius r = R s . The tumor aggressiveness is modeled by a positive tumor aggressiveness parameter μ . It is shown that there exist a positive integer m ∗ ∗ ∈ R and a sequence of μ m , such that for each μ m ( m > m ∗ ∗ ) , symmetry-breaking solutions bifurcate from the radially symmetric stationary solutions.

This paper deals with a parabolic system coupled via nonlocal sources, subjecting to positive Dirichlet boundary value conditions. By using the super-, sub-solution methods and techniques, and piecewise functions, the blow-up criteria and global boundedness of nonnegative solutions are determined. The results show the positive boundary value e 0 plays an important role in the case of blow-up.

This paper investigates the finite time blow-up of nonnegative solutions for a nonlinear diffusion system with a more complicated source term, which is a product of localized source, local source, and weight function, and complemented by homogeneous Dirichlet boundary conditions. The criteria are proposed to identify simultaneous and non-simultaneous blow-up solutions. Moreover, the related classification for the four parameters in the model is optimal and complete. The results extend those in Zhang and Yang [12].

In this paper we consider the weighted p-Laplacian \(-\mathrm{div}(|x|^\alpha |\nabla u|^{p-2}\nabla u)+ |x|^\gamma |u|^{q-1}u=0\) in \(B{\setminus }\{0\}\), where \(B\subset \mathbb {R}^N(N\ge 2)\) is the unit ball centered at the origin, \(\alpha \ge p-N\), \(\gamma >-N\) and \(q>p-1>0\). Classification of singularities for all positive solutions will be presented via two thresholds on \(\alpha \).

This work investigates a degenerate reaction–diffusion system with homogeneous Dirichlet boundary data. Using the self-similar form of function and super-solution sub-solution techniques, together with results from interactions among the multi-nonlinearities in the system, described using ten exponents, the global existence and blow-up criteria for nonnegative solutions are determined.

This paper deals with the large-time behaviour of solutions to the fast diffusive Newtonian filtration equations coupled via the nonlinear boundary sources. A result of Fujita type is obtained by constructing various kinds of upper and lower solutions. In particular, it is shown that the critical global existence curve and the critical Fujita curve concide for the multi-dimensional system. This is quite different from the known results obtained in Wang, Zhou and Lou [‘Critical exponents for porous medium systems coupled via nonlinear boundary flux’, Nonlinear Anal.7(1) (2009), 2134–2140] for the corresponding one-dimensional problem.

This paper considers the finite time blow-up of nonnegative solutions for a nonlinear diffusion system with a more complicated source term, which is a product of localized source, local source, weight function, and complemented by homogeneous Dirichlet boundary conditions. It is proved that there exists initial data such that simultaneous or non-simultaneous blow-up occur. Moreover, the related classifications for the four parameters in the model is optimal and complete. The results extend those in Zhang and Yang (J. Appl. Math. Comput. 32:429–441, 2010).

This paper deals with a degenerate parabolic system with nonlocal sources. It is prove that if pc = p1q1 − (m − p2)(n − q2) 0, there are solutions that blow-up and others that are global. When pc = 0, we show that if the domain is sufficiently small, every nonnegative solution is global while if the domain large enough that is, if it contains a sufficiently large ball, there is no global solution.

In this paper, we establish the blow-up theorems of Fujita type for a class of homogeneous Neumann exterior problems of quasilinear convection–diffusion equations. The critical Fujita exponents are determined and it is shown that the exponents belong to the blow-up case under any nontrivial initial data. Copyright © 2010 John Wiley & Sons, Ltd.

This paper deals with the large-time behaviour of solutions to the exterior problem of the non-Newtonian filtration equation with first-order term and nonlinear boundary source. In particular, the critical global exponent and the critical Fujita exponent are determined or estimated. An interesting phenomenon is shown: there exists a threshold value for the coefficient of the first-order term such that 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 value.

This paper is concerned with the porous medium equations for coupling via nonlinear boundary flux; we obtain the critical global existence curve and the critical Fujita curve for the problem, considered by constructing the self-similar supersolutions and subsolutions.

In this paper we consider a singular diffusion equation arising in phase transition and investigate its self-similar entropy solutions with jump hypersurfaces. The existence, nonexistence and uniqueness theorems of such solutions are established. We also discuss some properties of this kind of solutions including monotonicity and asymptotic behavior.

This paper is concerned with the large time behavior of solutions to two types of nonlinear diffusion equations with nonlinear boundary sources on the exterior domain of the unit ball. We are interested in the critical global exponent q 0 and the critical Fujita exponent q c for the problems considered, and show that q 0 = q c for the multi-dimensional porous medium equation and non-Newtonian filtration equation with nonlinear boundary sources. This is quite different from the known results that q 0 q c for the one-dimensional case.

This paper is concerned with large time behavior of solutions to the homogeneous Neumann problem of the non-Newtonian filtration equation. It is shown that the critical Fujita exponent for the problem considered is determined not only by the spatial dimension and the nonlinearity exponent, but also by the coefficient k of the first-order term. In fact, we show that there exist two thresholds k∞ and k1 on the coefficient k of the first-order term, and the critical Fujita exponent is a finite number when k is between k∞ and k1, while the critical exponent does not exist when k⩽k∞ or k⩾k1. Copyright © 2007 John Wiley & Sons, Ltd.

The paper concerns the existence theorem of weak solutions for the initial-boundary value problem of a nonlinear diffusion equation with convection. The equation may be regarded as a generalized non-Newtonian polytropic filtration equation. By doing the necessary BV estimate and other estimates for approximating solutions, we establish the existence theorem.

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.

In this paper, we establish the blow-up theorems of the Fujita type for a class of homogeneous Neumann problems of quasilinear equations with convection terms. The critical Fujita exponents are determined and it is shown that the exponents belong to the blow-up case under any nontrivial initial data. An interesting phenomenon is exploited such that the critical Fujita exponent even could be infinite for the model considered in the paper owing to the effect of convection.

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.

This paper is concerned with the p-Laplacian equation with singular sources which is allowed to change sign in a ball. The singularity may occur at the zero points of solutions and on the boundary. Using the upper and lower solutions method, we establish the existence of positive radial solutions for the problem considered. Copyright © 2006 John Wiley & Sons, Ltd.

In this paper, we consider the initial-boundary value problem of a nonlinear parabolic equation with double degeneracy, and establish the existence and uniqueness theorems of renormalized solutions which are stronger than $BV$ solutions.

In this paper, we study the similar entropy solutions of the singular diffusion equation, @?u@?t=@?@?x@j(@?u@?x), with @j(s)=s/1+s^2. These kinds of solutions have nonvertical jump lines. We establish the existence and uniqueness and also discuss some properties of these kinds of solutions.

This paper is concerned with a class of degenerate diffusion equations subject to mixed boundary conditions. Under some structure conditions, we discuss the blow-up property of local solutions and estimate the bounds of “blow-up time.”

This paper concerns with a parabolic system coupled via nonlocal sources, subjecting to homogeneous Dirichlet boundary condition. The main aim of this paper is to study conditions on the global existence and finite time blowup of solutions. By using the super- and sub-solution techniques, the critical exponent of the system is determined. Furthermore, the related classification for the parameters in the model is optimal and complete.

This paper investigates a compressible reactive gas model with homogeneous Dirichlet boundary conditions. Under the parameters and the initial data satisfying some conditions, we prove that the solutions have global blow-up, and the blow-up rate is uniform in all compact subsets of the domain. Moreover, the blow-up rates of |u(t)|∞ and |v(t)|∞ are precisely determined. MSC: 35K05; 35K55; 35D55

We prove the uniqueness of solutions to the initial boundary value problem of a viscous diffusion equation by means of a regularizing technique based on elliptic operators.