where D⊂RN (N≥ 2) is a bounded domain with smooth boundary ∂D. By constructing auxiliary functions and using maximum principles, the sufficient conditions for the existence of a global solution, an upper estimate of the global solution, the sufficient conditions for the existence of a blow-up solution, an upper bound for ‘blow-up time’, and an upper estimate of ‘blow-up rate’ are specified unde...

Abstract In this paper, we investigate the blow-up properties of a quasilinear reaction-diffusion system with nonlocal nonlinear sources and weighted nonlocal Dirichlet boundary conditions. The critical exponent is determined under various situations of the weight functions. It is observed that the boundary weight functions play an important role in determining the blow-up conditions. In additi...

In this work, we study the blow-up and global solutions for a quasilinear reaction–diffusion equation with a gradient term and nonlinear boundary condition:      (g(u)) t = ∆u + f (x, u, |∇u| 2 , t) in D × (0, T), ∂u ∂n = r(u) on ∂D × (0, T), u(x, 0) = u 0 (x) > 0 in D, where D ⊂ R N is a bounded domain with smooth boundary ∂D. Through constructing suitable auxiliary functions and using ma...

In this paper, we show existence, uniqueness and exact asymptotic behavior of solutions near the boundary to a class of semilinear elliptic equations −∆u = λg(u)− b(x)f(u) in Ω, where λ is a real number, b(x) > 0 in Ω and vanishes on ∂Ω. The special feature is to consider g(u) and f(u) to be regularly varying at infinity and b(x) is vanishing on the boundary with a more general rate function. T...

In this paper we consider a kind of higher-order evolution equation as^{kt^{k} + ^{k&minus1}u/t^{k&minus1} +• • •+ut &minus{delta}u= f (u, {delta}u,x). For this equation, we investigate nonglobal solution, blow-up in finite time and instantaneous blow-up under some assumption on k, f and initial data. In this paper we employ the Test function method, the eneralized convexity method an...

where p > 2 and is a bounded domain in Rn (n≥ 2) with smooth boundary ∂ . By introducing some appropriate auxiliary functions and technically using maximum principles, we establish conditions to guarantee that the solution blows up in some finite time or remains global. In addition, the upper estimates of blow-up rate and global solution are specified. We also obtain an upper bound of blow-up t...

We rigorously construct radial H solutions to the 3d cubic focusing NLS equation i∂tψ + ∆ψ + 2|ψ|ψ = 0 that blow-up along a contracting sphere. With blow-up time set to t = 0, the solutions concentrate on a sphere at radius ∼ t but focus towards this sphere at the faster rate ∼ t. Such dynamics were originally proposed heuristically by Degtyarev-Zakharov-Rudakov [2] in 1975 and independently la...

for the polynomial planar vector fields with a hyperbolic or nilpotent critical point at the origin, the monodromy problem has been solved, but for the strongly degenerate critical points this problem is still open. when the critical point is monodromic, the stability problem or the center- focus problem is an open problem too. in this paper we will consider the polynomial planar vector fields ...