Global Blow-up Research Articles

Uniform blow-up rate for parabolic equations with a weighted nonlocal nonlinear source

In this paper, the blow-up rate of solutions of semi-linear reaction–diffusion equations with a more complicated source term, which is a product of nonlocal (or localized) source and weight function a ( x ) , is investigated. It is proved that the solutions have global blow-up, and that the rates of blow-up are uniform in all compact subsets of the domain. Furthermore, the blow-up rate of | u ( t ) | ∞ is precisely determined.

Cauchy–Goursat Problem for One-Dimensional Semilinear Wave Equations

We consider the initial-characteristic problem for nonlinear wave equations with positive power nonlinearity source term. Depending on the power of nonlinearity, we investigate the problem on a global existence and blow-up of solutions of initial-characteristic problem. The question on local solvability of the problem is also considered.

A class of nonlocal and degenerate quasilinear parabolic system not in divergence form

In this paper, we investigate the positive solution of the nonlocal and degenerate quasilinear parabolic system not in divergence form u t = f ( u ) ( Δ u + a ∫ Ω v d x ) , v t = g ( v ) ( Δ v + b ∫ Ω u d x ) with null Dirichlet boundary conditions. The blow-up and global existence criteria are obtained. Furthermore, we prove that under certain conditions the solutions have global blow-up.

The Porous Medium Equation With Blowing Up Boundary Data

Abstract We study the blow-up set of a solution u(x, t) to the porous medium equation, ut = Δ(um), in Ω × (0, T) for two different boundary conditions, u(x, t) = f(x)/(T - t) or (x, t) = g(x)/(T - t) on ӘΩ × (0, T). Here m > 1 and Ω is a bounded smooth domain in ℝN. We establish point-wise and integral conditions on f and g respectively in order to obtain regional or global blow-up.

A cross-constrained variational problem for the generalized Davey–Stewartson system

AbstractWe study the sharp threshold for blow-up and global existence and the instability of standing wave eiωtuω(x) for the Davey–Stewartson systemin ℝ3, where uω is a ground state. By constructing a type of cross-constrained variational problem and establishing so-called cross-invariant manifolds of the evolution flow, we derive a sharp criterion for global existence and blow-up of the solutions to (DS), which can be used to show that there exist blow-up solutions of (DS) arbitrarily close to the standing waves.

Global existence and blow-up analysis for some degenerate and quasilinear parabolic systems

This paper deals with positive solutions of some degenerate and quasilinear parabolic systems not in divergence form: u1t = f1(u2)(�u1 + a1u1), � � � ,u(n 1)t = fn 1(un)(�un 1 + an 1un 1), unt = fn(u1)(�un+anun) with homogenous Dirichlet boundary condition and posi- tive initial condition, where ai (i = 1,2, ��� ,n) are positive constants and fi (i = 1,2, ��� ,n) satisfy some conditions. The local existence and uniqueness of classical solution are proved. Moreover, it will be proved that: (i) when min{a1, � � � , an} ≤ �1 then there exists global positive classical solution, and all positive classical solutions can not blow up in finite time in the meaning of maximum norm; (ii) when min{a1, � � � , an} > �1, and the initial datum (u10, � � � , un0) satisfies some assumptions, then the positive classical solution is unique and blows up in finite time, where �1 is the first eigenvalue of −� in with homogeneous Dirichlet boundary condition.

Sliding Vector Fields via Slow--Fast Systems

This paper concerns differential equation systems on $\mathbb R^n$ with discontinuous right--hand sides. We deal with non-smooth vector fields in $\mathbb R^n$ having a codimension-one submanifold $M$ as its discontinuity set. After a regularization of a such system and a global blow-up we are able to bring out some results that bridge the space between discontinuous systems and singularly perturbed smooth systems.

Global solution for a generalized Boussinesq equation

This paper considers the existence and uniqueness of the solution to the Cauchy problem for a class of Boussinesq equation, proves the global existence and finite time blowup of the solution to the problem by the potential well method.

A Reaction-diffusion system with nonlinear absorption terms and boundary flux

This paper deals with a reaction-diffusion system with nonlinear absorption terms and boundary flux. As results of interactions among the six nonlinear terms in the system, some sufficient conditions on global existence and finite time blow-up of the solutions are described via all the six nonlinear exponents appearing in the six nonlinear terms. In addition, we also show the influence of the coefficients of the absorption terms as well as the geometry of the domain to the global existence and finite time blow-up of the solutions for some cases. At last, some numerical results are given.

Threshold of damped critical non-linear Schrödinger equation with fourth-order dispersion

This article is concerned with a damped critical non-linear Schrödinger equation with fourth-order dispersion which models propagation of fibre arrays in non-linear Kerr media. We analyse the effect of the damping force on the solution for the system and prove that there exists a threshold value of the damping parameter for global existence and blowup of the system. Furthermore, we estimate the threshold value.

Three types of self-similar blow-up for the fourth-order [formula omitted]-Laplacian equation with source

Self-similar blow-up behaviour for the fourth-order quasilinear p -Laplacian equation with source, u t = − ( | u x x | n u x x ) x x + | u | p − 1 u in R × R + , where n > 0 , p > 1 , is studied. Using variational setting for p = n + 1 and branching techniques for p ⁄ = n + 1 , finite and countable families of blow-up patterns of the self-similar form u S ( x , t ) = ( T − t ) − 1 p − 1 f ( y ) , where y = x / ( T − t ) β , β = − p − ( n + 1 ) 2 ( n + 2 ) ( p − 1 ) , are described by an analytic-numerical approach. Three parameter ranges: p = n + 1 (regional), p > n + 1 (single point), and 1 < p < n + 1 (global blow-up) are studied. This blow-up model is motivated by the second-order reaction-diffusion counterpart u t = ( | u x | n u x ) x + u p ( u ≥ 0 ) that was studied in the middle of the 1980s, while first results on blow-up of solutions were established by Tsutsumi in 1972.

Global existence and blow-up for degenerate and singular parabolic system with localized sources

This paper deals with the blow-up properties of the solution to degenerate and singular parabolic system with localized sources and homogeneous Dirichlet boundary conditions. The existence of a unique classical nonnegative solution is established and the sufficient conditions for the solution exists globally and blows up in finite time are obtained. Furthermore, under certain conditions, it is proved that the blow-up set of the solution is the whole domain.

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.

Global Dynamics of Blow-up Profiles in One-dimensional Reaction Diffusion Equations

We consider reaction diffusion equations of the prototype form u t = u xx + λ u + |u| p-1 u on the interval 0 1 and λ > m 2. We study the global blow-up dynamics in the m-dimensional fast unstable manifold of the trivial equilibrium u ≡ 0. In particular, sign-changing solutions are included. Specifically, we find initial conditions such that the blow-up profile u(t, x) at blow-up time t = T possesses m + 1 intervals of strict monotonicity with prescribed extremal values u 1, . . . ,u m . Since u k = ± ∞ at blow-up time t = T, for some k, this exhausts the dimensional possibilities of trajectories in the m-dimensional fast unstable manifold. Alternatively, we can prescribe the locations x = x 1, . . . ,x m of the extrema, at blow-up time, up to a one-dimensional constraint. The proofs are based on an elementary Brouwer degree argument for maps which encode the shapes of solution profiles via their extremal values and extremal locations, respectively. Even in the linear case, such an “interpolation of shape” was not known to us. Our blow-up result generalizes earlier work by Chen and Matano (1989), J. Diff. Eq. 78, 160–190, and Merle (1992), Commun. Pure Appl. Math. 45(3), 263–300 on multi-point blow-up for positive solutions, which were not constrained to possess global extensions for all negative times. Our results are based on continuity of the blow-up time, as proved by Merle (1992), Commun. Pure Appl. Math. 45(3), 263–300, and Quittner (2003), Houston J. Math. 29(3), 757–799, and on a refined variant of Merle’s continuity of the blow-up profile, as addressed in the companion paper Matano and Fiedler (2007) (in preparation).

Global blow-up for a heat system with localized sources and absorptions

In this paper there are established the global existence and finite time blow-up results of nonnegative solution for the following parabolic system $$\begin{array}{*{20}c} {u_t = \Delta u + v^p (x_0 ,t) - au^r , x \in \Omega , t > 0,} \\ {v_t = \Delta v + u^q (x_0 ,t) - bv^s , x \in \Omega , t > 0,} \\ \end{array} $$ subject to homogeneous Dirichlet conditions and nonnegative initial data, where x0 ∈ Ω is a fixed point, p, q, r, s ≥ 1 and a, b > 0 are constants. In the situation when nonnegative solution (u, υ) of the above problem blows up in finite time, it is showed that the blow-up is global and this differs from the local sources case. Moreover, for the special case r = s = 1, $$\begin{array}{*{20}c} {\mathop {\lim }\limits_{t \to T*} (T* - t)^{\frac{{p + 1}}{{pq - 1}}} u(x,t) = (p + 1)^{\frac{1}{{pq - 1}}} (q + 1)^{\frac{p}{{pq - 1}}} (pq - 1)^{ - \frac{{p + 1}}{{pq - 1}}} ,} \\ {\mathop {\lim }\limits_{t \to T*} (T* - t)^{\frac{{q + 1}}{{pq - 1}}} v(x,t) = (p + 1)^{\frac{q}{{pq - 1}}} (q + 1)^{\frac{1}{{pq - 1}}} (pq - 1)^{ - \frac{{q + 1}}{{pq - 1}}} ,} \\ \end{array} $$ are obtained uniformly on compact subsets of Ω, where T* is the blow-up time.

Global solutions and blow-up problems for a nonlinear degenerate parabolic system coupled via nonlocal sources

This paper concerns with a nonlinear degenerate 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/or blow-up in finite time of solutions, and give the estimates of blow-up rates of blow-up solutions.

Regional Blow-up for a Doubly Nonlinear Parabolic Equation with a Nonlinear Boundary Condition

In this work, positive solutions to a doubly nonlinear parabolic equation with a nonlinear boundary condition are considered. We study the problem $${\left\{\begin{array}{ll} (u^m)_t = (|u_x|^{r-1}u_x)_x, & (x,t)\in (0,L)\times(0,T_{\rm max}),\\ u_x(0,t)=0\;, \quad|u_x|^{r-1}u_x(L,t)=u^{\alpha}(L,t), & t\in(0,T_{\rm max}),\\ u(x,0)=\varphi(x), & x\in[0,L],\;\end{array}\right.}$$ where 0 < m, r, α < ∞ are parameters. It is known that for some values of the parameters there are solutions that blow up in finite time. We determine in terms of α ,m, r the blow-up sets for these solutions. We prove that single point blow-up occurs if max{m, r} < α, global blow-up appears for the range of parameters 0 < m < α < r and regional blow-up takes place if 0 < m < α = r and $$L > L^{*}\equiv \frac{r+1}{r-m}$$ . In this case the blow-up set consists of the interval $${[L-L^\ast,L]\subset (0,L]}$$ .

Uniform blow-up rate for diffusion equations with nonlocal nonlinear source

In this paper, we investigate the blow-up rate of solutions of diffusion equations with nonlocal nonlinear reaction terms. For large classes of equations, we prove that the solutions have global blow-up and that the rate of blow-up is uniform in all compact subsets of the domain. In each case, the blow-up rate of | u ( t ) | ∞ is precisely determined.

Asymptotic blow-up behavior for a nonlocal degenerate parabolic equation

In this paper, we investigate the positive solution of nonlinear degenerate equation u t = f ( u ) ( Δ u + a ( x ) ∫ Ω u d x ) with Dirichlet boundary condition. The blow-up criteria is obtained. Furthermore, we prove that under certain conditions, the solutions have global blow-up. When f ( u ) = u p , 0 < p ⩽ 1 , we gained blow-up rate estimate.

ON THE LAPLACE EQUATION WITH NON-LINEAR DYNAMICAL BOUNDARY CONDITIONS

The main part of the paper deals with local existence and global existence versus blow-up for solutions of the Laplace equation in bounded domains with a non-linear dynamical boundary condition. More precisely, we study the problem consisting in: (1) the Laplace equation in $(0, \infty) \times \Omega$; (2) a homogeneous Dirichlet condition $(0, \infty) \times \Gamma_0$; (3) the dynamical boundary condition $ \frac {\partial u}{\partial \nu} = - |u_t|^{m-2} u_t + |u|^{p - 2} u$ on $(0, \infty) \times \Gamma_1$; (4) the initial condition $u(0, x) = u_0 (x)$ on $\partial \Omega$. Here $\Omega$ is a regular and bounded domain in $\mathbb{R}^n$, with $n \ge 1$, and $\Gamma_0$ and $\Gamma_1$ endow a measurable partition of $\partial \Omega$. Moreover, $m&gt;1$, $2 \le p &lt; r$, where $r = 2 (n - 1) / (n - 2)$ when $n \ge 3$, $r = \infty$ when $n = 1,2$, and $u_0 \in H^{1/2} (\partial \Omega)$, $u_0 = 0$ on $\Gamma_0$.The final part of the paper deals with a refinement of a global non-existence result by Levine, Park and Serrin, which is applied to the previous problem.