Global solution and blow-up of a semilinear heat equation with logarithmic nonlinearity

We study the initial boundary value problem of a semilinear heat equation with logarithmic nonlinearity. By using the logarithmic Sobolev inequality and a family of potential wells, we obtain the existence of global solution and blow-up at +∞ under some suitable conditions. On the other hand, the results for decay estimates of the global solutions are also given. Our result in this paper means that the polynomial nonlinearity is a critical condition of blow-up in finite time for the solutions of semilinear heat equations.     

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.     

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.     

Regularity for a fourth-order critical equation with gradient nonlinearity

Given Ω a smooth bounded domain of Rn, n?3, we consider functions that are weak solutions to the equation

     

Global existence and blow-up of solutions for the heat equation with exponential nonlinearity

We are concerned with the existence of global in time solution for a semilinear heat equation with exponential nonlinearity

(P)$\{\begin{array}{cc}{?}_{t}u=\mathrm{\Delta }u+e^u,\hfill & x\in {\mathbf{R}}^N,t>0,\hfill \\ u(x,0)=u_0(x),\hfill & x\in {\mathbf{R}}^N,\hfill \end{array}$

where $u_0$ is a continuous initial function. In this paper, we consider the case where $u_0$ decays to ?∞ at space infinity, and study the optimal decay bound classifying the existence of global in time solutions and blowing up solutions for (P). In particular, we point out that the optimal decay bound for $u_0$ is related to the decay rate of forward self-similar solutions of ${?}_{t}u=\mathrm{\Delta }u+e^u$.     

Convergence to a singular steady state of a parabolic equation with gradient blow-up

We study solutions of a parabolic equation which are bounded but whose spatial derivatives blow up in finite time. We establish results on the behavior on the lateral boundary where the singularity occurs and on the rate of convergence to a singular steady state.     

Numerical solution of a diffusion equation with a reproducing nonlinearity

Summary We study the Faedo-Galerkin approximations of the Burger Equation, which we write as an operator equation of the typeu _t+Au+Nu=0 in a Hilbert spaceH. We show that the nonlinear operatorN is finitely reproducing relative to the orthonormal sequence {u _i} generated byAu=u and study the numerical behavior of the approximations.

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Zusammenfassung Wir untersuchen die Faedo-Galerkin Approximationen der Burger Gleichung, die wir als Operatorgleichung des Typesu _t+Au+Nu=0, in einem Hilbertraum auffassen. Wir zeigen, daß der nichtlineare OperatorN, endlich reproduzierend bezüglich einer Orthonormalfolgeu _i ist, die durchAu=u erzeugt wird und untersuchen das numerische Verhalten der Approximationen.

     

On the nonextendable solution and blow-up of the solution of the one-dimensional equation of ion-sound waves in a plasma

The initial boundary-value problem for the equation of ion-sound waves in a plasma is studied. A theorem on the nonextendable solution is proved. Sufficient conditions for the blow-up of the solution in finite time and the upper bound for the blow-up time are obtained using the method of test functions.     

Rate estimates of gradient blowup for a heat equation with exponential nonlinearity

We consider a one-dimensional semilinear parabolic equation , for which the spatial derivative of solutions becomes unbounded in finite time while the solutions themselves remain bounded. We establish estimates of blowup rate upper and lower bounds. We prove that in this case the blowup rate does not match the one obtained by the rescaling method.     

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.     

Asymptotics for the blow-up boundary solution of the logistic equation with absorption

Let $\text{Ω}$ be a smooth bounded domain in $\text{R}{}^{\mathrm{N}}$. Assume that f?0 is a C¹-function on [0,∞) such that f(u)/u is increasing on (0,+∞). Let a be a real number and let b?0, b?0 be a continuous function such that b≡0 on $\text{?Ω}$. The purpose of this Note is to establish the asymptotic behaviour of the unique positive solution of the logistic problem Δu+au=b(x)f(u) in $\text{Ω}$, subject to the singular boundary condition u(x)→+∞ as $\text{dist}\text{(x,?Ω)→0}$. Our analysis is based on the Karamata regular variation theory. To cite this article: F.-C. Cîrstea, V. R?dulescu, C. R. Acad. Sci. Paris, Ser. I 336 (2003).     

On the behaviour of blow-up interfaces for an inhomogeneous filtration equation

We study the asymptotic behaviour of blow-up interfaces of thesolutions to the one-dimensional nonlinear filtration equationin inhomogeneous media where m>1 isa constant and (x) = |x|^– (for |x| 1, with > 2) isa bounded, positive, smooth, and symmetric function. The initialdata are assumed to be smooth, bounded, compactly supported,symmetric, and monotone. It is known that due to the fast decayof the density (x) as |x| the support of the solution increasesunboundedly in a finite time T. We prove that as tT^– theinterface behaves like O((T – t)^–b), where the exponentb > 0 (which depends on m and only) is given by a uniqueself-similar solution of the second kind satisfying the equation|x|^– u_t = (u^m)_xx. The corresponding rescaled profilesalso converge. We establish the stability of the self-similarsolution of the second kind for the exponential density (x)=e^–|x|for |x| 1. We give a formal asymptotic analysis of the blow-upbehaviour for the non-self-similar density (x) = e^–|x|2.Several exact self-similar solutions and their correspondingasymptotics are constructed.     

On the blow-up of solutions to anisotropic parabolic equations with variable nonlinearity

The aim of this paper is to establish sufficient conditions of the finite time blow-up in solutions of the homogeneous Dirichlet problem for the anisotropic parabolic equations with variable nonlinearity $ u_t = \sum\nolimits_{i = 1}^n {D_i (a_i (x,t)|D_i u|^{p^i (x) - 2} D_i u) + \sum\nolimits_{i = 1}^K {b_i (x,t)|u|^{\sigma _i (x,t) - 2} u} } $ u_t = \sum\nolimits_{i = 1}^n {D_i (a_i (x,t)|D_i u|^{p^i (x) - 2} D_i u) + \sum\nolimits_{i = 1}^K {b_i (x,t)|u|^{\sigma _i (x,t) - 2} u} } . Two different cases are studied. In the first case a _i ≡ a _i (x), p _i ≡ 2, σ _i ≡ σ _i (x, t), and b _i (x, t) ≥ 0. We show that in this case every solution corresponding to a “large” initial function blows up in finite time if there exists at least one j for which min σ _j (x, t) > 2 and either b _j > 0, or b _j (x, t) ≥ 0 and Σ_π b _j ^−ρ(t)(x, t) dx < ∞ with some σ(t) > 0 depending on σ _j . In the case of the quasilinear equation with the exponents p _i and σ _i depending only on x, we show that the solutions may blow up if min σ _i ≥ max p _i , b _i ≥ 0, and there exists at least one j for which min σ _j > max p _j and b _j > 0. We extend these results to a semilinear equation with nonlocal forcing terms and quasilinear equations which combine the absorption (b _i ≤ 0) and reaction terms.     

含梯度项的椭圆方程组的边界爆破解

研究了含梯度项的椭圆方程组的边界爆破解的性质,其中权函数a(x),b(x)为正并且满足一定的条件.利用上下解的方法及比较原则证明了正解的存在性与唯一性,并得到了边界爆破速率的估计.     

On a singular parabolic p-biharmonic equation with logarithmic nonlinearity

This paper is concerned with the well-posedness and asymptotic behavior of Dirichlet initial boundary value problem for a singular parabolic p-biharmonic equation with logarithmic nonlinearity. We establish the local solvability by the technique of cut-off combining with the methods of Faedo–Galerkin approximation and multiplier. Meantime, by virtue of the family of potential wells, we use the technique of modified differential inequality and improved logarithmic Sobolev inequality to obtain the global solvability, infinite and finite time blow-up phenomena, and derive the upper bound of blow-up time as well as the estimate of blow-up rate. Furthermore, the results of blow-up with arbitrary initial energy and extinction phenomena are presented.     

On the supercritical KdV equation with time-oscillating nonlinearity

For the initial value problem (IVP) associated to the generalized Korteweg–de Vries (gKdV) equation with supercritical nonlinearity, $$u_{t}+\partial_x^3u+\partial_x(u^{k+1}) =0,\qquad k\geq 5,$$ numerical evidence [3] shows that, there are initial data ${\phi\in H^1(\mathbb{R})}$ such that the corresponding solution may blow-up in finite time. Also, with the evidence from numerical simulation [1, 18], it has been claimed that a periodic time dependent coefficient in the nonlinearity would disturb the blow-up solution, either accelerating or delaying it. In this work, we investigate the IVP associated to the gKdV equation $$u_{t}+\partial_x^3u+g(\omega t)\partial_x(u^{k+1}) =0,$$ where g is a periodic function and ${k\geq 5}$ is an integer. We prove that, for given initial data ${\phi \in H^1(\mathbb{R})}$ , as ${|\omega|\to \infty}$ , the solution ${u_{\omega} }$ converges to the solution U of the initial value problem associated to $$U_{t}+\partial_x^3U+m(g)\partial_x(U^{k+1}) =0,$$ with the same initial data, where m(g) is the average of the periodic function g. Moreover, if the solution U is global and satisfies ${\|U\|_{L_x^{5}L_t^{10}}<\infty}$ , then we prove that the solution ${u_{\omega} }$ is also global provided ${|\omega|}$ is sufficiently large.     

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.     

Finite time blow-up and global existence of weak solutions for pseudo-parabolic equation with exponential nonlinearity

This paper is concerned with the initial boundary value problem of a class of pseudo-parabolic equation $u_t - \triangle u - \triangle u_t + u = f(u)$ with an exponential nonlinearity. The eigenfunction method and the Galerkin method are used to prove the blow-up, the local existence and the global existence of weak solutions. Moreover, we also obtain other properties of weak solutions by the eigenfunction method.     

1-Soliton solution of the Klein-Gordon-Zakharov equation with power law nonlinearity

This paper obtains the 1-soliton solution of the Klein-Gordon-Zakharov equation with power law nonlinearity. The solutions are obtained both in (1+1) and (1+2) dimensions. The solitary wave Ansatz method is applied to obtain the solution. The numerical simulations are included that supports the analysis.