The parabolic logistic equation with blow-up initial and boundary values

In this article, we investigate the parabolic logistic equation with blow-up initial and boundary values $${u_t} - \Delta u = a(x,t)u - b(x,t){u^p}in\Omega \times (0,T),$$ $$u = \infty on\partial \Omega \times (0,T) \cup \overline \Omega \times \{ 0\} ,$$ where ?? is a smooth bounded domain, T > 0 and p > 1 are constants, and a and b are continuous functions, b > 0 in ?? × [0, T) and b(x, T) ?? 0. We study the existence and uniqueness of positive solutions and their asymptotic behavior near the parabolic boundary. We show that under the extra condition that $b(x,t) \ge c{(T - t)^\theta }d{(x,\partial \Omega )^\beta } on \Omega \times \left[ {0,T} \right)$ for some constants c > 0, ?? > 0, and ?? > ?2, such a solution stays bounded in any compact subset of ?? as t increases to T, and hence solves the equation up to t = T.     

Uniqueness of the blow-up boundary solution of logistic equations with absorbtion

Let $\text{Ω}$ be a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$. Assume f∈C¹[0,∞) is a non-negative function such that f(u)/u is increasing on (0,∞). Let a be a real number and let b?0, $\text{b}\text{/≡0}$ be a continuous function such that b≡0 on $\text{?Ω}$. We study the logistic equation Δu+au=b(x)f(u) in $\text{Ω}$. The special feature of this work is the uniqueness of positive solutions blowing-up on $\text{?Ω}$, in a general setting that arises in probability theory. To cite this article: F.-C. C??rstea, V. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 335 (2002) 447–452.     

The blow-up rate and uniqueness of large solution for a porous media logistic equation

In this paper we ascertain the exact blow-up rate of the large solutions of a class of sublinear elliptic problems of a logistic type related to the porous media equation, from which we can obtain the uniqueness of the solution. The weight function in front of the nonlinearity vanishes on the boundary of the underlying domain with a general decay rate which can be approximated by a distance function.     

Asymptotics of the solution of Laplace’s equation with mixed boundary conditions

A uniform asymptotic expansion of the solution of a two-dimensional elliptic problem with mixed boundary conditions is found. A physical application of the result is discussed.     

Boundary behavior of solutions for the degenerate logistic type elliptic problem with boundary blow-up

By the Karamata regular variation theory and constructing comparison function, we show the exact asymptotic behavior of solutions for the degenerate logistic type elliptic problem with boundary blow-up.     

On decay and blow-up of the solution for a viscoelastic wave equation with boundary damping and source terms

In this paper we consider the decay and blow-up properties of a viscoelastic wave equation with boundary damping and source terms. We first extend the decay result (for the case of linear damping) obtained by Lu et al. (On a viscoelastic equation with nonlinear boundary damping and source terms: Global existence and decay of the solution, Nonlinear Analysis: Real World Applications 12 (1) (2011), 295-303) to the nonlinear damping case under weaker assumption on the relaxation function g(t). Then, we give an exponential decay result without the relation between g^′(t) and g(t) for the linear damping case, provided that ‖g‖_L¹(0,∞) is small enough. Finally, we establish two blow-up results: one is for certain solutions with nonpositive initial energy as well as positive initial energy for both the linear and nonlinear damping cases, the other is for certain solutions with arbitrarily positive initial energy for the linear damping case.     

Diffusive logistic equation with non-linear boundary conditions

We analyze the solutions of a population model with diffusion and logistic growth. In particular, we focus our study on a population living in a patch, Ω⊆Rn with n?1, that satisfies a certain non-linear boundary condition and on its survival when constant yield harvesting is introduced. We establish our existence results by the method of sub-super solutions.     

On blow-up of solution of nonhomogeneous polytropic equation with source

In this paper we discuss the blow-up property of the positive solutions of the mixed problem for a one-dimensional polytropic filtration equation with source.     

Existence of boundary blow-up solutions for a quasilinear elliptic equation

In this paper, the existence of solution for a class of quasilinear elliptic problem div(|? u| ^p?2 ? u)=a(x)f(u), u≥0 in Ω=B (the unit ball), with the boundary blow-up condition u| _?Ω=+∞ is established, where a(x)∈C(Ω) blows up on ?Ω,p>1 and f is assumed to satisfy (f ₁) and (f ₂). The results are obtained by using sub-supersolution methods.     

Remarks on blow-up behavior for a nonlinear diffusion equation with Neumann boundary conditions

We establish the blow-up rate for the solution of a nonlinear diffusion equation , subject to Neumann boundary conditions .

     

Global blow-up for a localized nonlinear parabolic equation with a nonlocal boundary condition

This paper deals with the blow-up properties of positive solutions to a nonlinear parabolic equation with a localized reaction source and a nonlocal boundary condition. Under certain conditions, the blowup criteria is established. Furthermore, when f(u)=up, 0<p?1, the global blowup behavior is shown, and the blowup rate estimates are also obtained.     

On boundary blow-up problems for the complex Monge-Ampère equation

We prove the regularity for some complex Monge-Ampère equations with boundary data equal to .

     

On the blow-up behavior of a nonlinear parabolic equation with periodic boundary conditions

Using ideas arising in the works of LeJan and Sznitman and Mattingly and Sinai on their study of the Navier–Stokes equations, we investigate the blow-up behavior of a nonlinear parabolic equation subject to periodic boundary conditions.     

Existence and blow-up of solutions for a porous medium equation with nonlocal boundary condition

In this article, a porous medium equation with nonlocal boundary condition and a localized source is studied. The results of the existence of global solutions or blow-up of solutions are given. The blow-up rate estimates are also obtained under some conditions.     

On the blow-up of the solution of an equation with a gradient nonlinearity

We continue the study of a nonlinear third-order equation of the Hamilton-Jacobi type. For this equation, we consider an initial-boundary value problem in a bounded domain with smooth boundary and prove the local solvability in the strong generalized sense; in addition, we derive sufficient conditions for the blow-up in finite time and sufficient conditions for the time-global solvability.     

Critical exponents and blow-up rate for a nonlinear diffusion equation with logarithmic boundary flux

In this paper, we establish the critical global existence exponent and the critical Fujita exponent for the nonlinear diffusion equation u_t=(log^σ(1+u)u_x)_x, in R₊×(0,+∞), subject to a logarithmic boundary flux , furthermore give the blow-up rate for the nonglobal solutions.     

On the blow-up of the solution of an equation related to the Hamilton-Jacobi equation

A new model three-dimensional third-order equation of Hamilton-Jacobi type is derived. For this equation, the initial boundary-value problem in a bounded domain with smooth boundary is studied and local solvability in the strong generalized sense is proved; in addition, sufficient conditions for the blow-up in finite time and sufficient conditions for global (in time) solvability are obtained.     

The uniqueness of blow-up solution for radially symmetric semilinear elliptic equation

In this paper we establish a blow up rate of the large positive solutions of the singular boundary value problem -Δu=λu-b(x)u^p,u_|∂Ω=+∞$-\mathrm{\Delta }u=\lambda u-b(x)u^p,u{|}_{\mathrm{\partial }\Omega }=+\infty$ with a ball domain and radially function b(x)$b(x)$. All previous results in the literature assumed the decay rate of b(x)$b(x)$ to be approximated by a distance function near the boundary ∂Ω$\mathrm{\partial }\Omega$. Obtaining the accurate blow up rate of solutions for general b(x)$b(x)$ requires more subtle mathematical analysis of the problem.