Uniform blow-up rate for compressible reactive gas model

The Dirichlet initial-boundary value problem of a compressible reactive gas model equation with a nonlocal nonlinear source term is investigated. Under certain conditions, it can be proven that the blow-up rate is uniform in all compact subsets of the domain, and the blow-up rate is irrelative to the exponent of the diffusion term, however, relative to the exponent of the nonlocal nonlinear source.     

BLOW-UP OF THE SOLUTIONS FOR THE INITIAL-BOUNDARY PROBLEMS OF THE NONLINEAR SCHR?DINGER EQUATIONS

Ｔｈｅｒｅａｒｅｍａｎｙｄｏｃｕｍｅｎｔｓｔｈａｔｄｉｓｃｕｓｓｂｌｏｗ＿ｕｐｏｆｔｈｅｓｏｌｕｔｉｏｎｓｆｏｒｔｈｅｉｎｉｔｉａｌ＿ｂｏｕｎｄａｒｙｖｌｕｅｐｒｏｂｌｅｍｓｏｆｔｈｅｎｏｎｌｉｎｅａｒＳｃｈｒ ｄｉｎｇｅｒｅｑｕａｔｉｏｎｓ,ｓｕｃｈａｓｐａｐｅｒ [1 ] .Ｂｕｔｔｈｅｐａｐｅｒ [1 ]ｈａｓｏｎｅｗｅａｋｐｏｉｎｔ,ｔｈａｔｉｓ,ｉｔｃａｎｎｏｔｕｎｉｔｅｄｉｓｃｕｓｓｉｏｎｆｏｒｂｌｏｗ＿ｕｐｏｆｔｈｅｓｏｌｕｔｉｏｎｓｆｏｒｔｈｅｉｎｉｔｉａｌ＿ｂｏｕｎｄａｒｙｖａｌｕｅｐｒｏｂｌ…     

Blow-up rate estimate for degenerate parabolic equation with nonlinear gradient term

Abstract In this paper, the blow-up rate is obtained for a porous medium equation with a nonlinear gradient term and a nonlinear boundary flux. By using a scaling method and regularity estimates of parabolic equations, the blow-up rate determined by the interaction between the diffusion and the boundary flux is obtained. Compared with previous results, the gradient term, whose exponent does not exceed two, does not affect the blow-up rate of the solutions.     

The pansystems view of prediction and blow-up of fluid

ＴＨＥＰＡＮＳＹＳＴＥＭＳＶＩＥＷＯＦＰＲＥＤＩＣＴＩＯＮＡＮＤＢＬＯＷ－ＵＰＯＦＦＬＵＩＤＯｕｙａｎｇＳｈｏｕ－ｃｈｅｎｇ（欧阳首承）（ＣｈｅｎｇｄｕＭｅｔｅｏｒｏｌｏｇｉｃａｌＩｎｓｔｉｔｕｔｅ,Ｃｈｅｎｇｄｕ６１００４１）（ＲｅｃｅｉｖｅｄＪａ...     

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 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 . In this case the blow-up set consists of the interval .     

A Note on Diffusion-Induced Blow-up

The goal of paper is to give a simpler proof and some extensions of a result of Weinberger [3] concerning diffusion induced blow-up. The result states that, for certain systems of two parabolic equations with equal diffusion and homogeneous Neumann boundary conditions, some blowing-up solutions exist, although the corresponding system of ODE’s has only global bounded solutions. To Professor Pavol Brunovsky, on the occasion of his 70th birthday.     

BLOW－UP AND DIE－OUT OF SOLUTIONS OF NONLINEAR PSEUDO-HYPERBOLIC EQUATIONS OF GENERALIZED NERVE CONDUCTION TYPE

ＢＬＯＷ－ＵＰＡＮＤＤＩＥ－ＯＵＴＯＦＳＯＬＵＴＩＯＮＳＯＦＮＯＮＬＩＮＥＡＲＰＳＥＵＤＯＨＹＰＥＲＢＯＬＩＣＥＱＵＡＴＩＯＮＳＯＦＧＥＮＥＲＡＬＩＺＥＤＮＥＲＶＥＣＯＮＤＵＣＴＩＯＮＴＹＰＥＷａｎｇＦａｎｂｉｎ（王凡彬）（ＲｅｃｅｉｖｅｄＪｕｎｅ２...     

ASYMPTOTIC NON-STABILITY AND BLOW-UP AT BOUNDARY FOR SOLUTIONS OF A FILTRATION EQUATION

For a class of nonlinear filtration equation with nonlinear second-third boundary value condition, it is shown that a priori boundary of the solution can be estimated and controlled by initial data and integral on the boundary of the region. The priori estimate of the solutions was established by iterative method. By using this estimate the solutions may blow-up on the boundary of the region and thus it may have asymptotic non-stability.     

Global existence and blow-up phenomena of classical solutions for the system of compressible adiabatic flow through porous media

By means of maximum principle for nonlinear hyperbolic systems, the results given by HSIAO Ling and D. Serre was improved for Cauchy problem of compressible adiabatic flow through porous media, and a complete result on the global existence and the blowup phenomena of classical solutions of these systems. These results show that the dissipation is strong enough to preserve the smoothness of ‘ small ‘ solution.     

Exact conditions of blow-up and global existence for the nonlinear wave equation with damping and source terms

This paper deals with the Cauchy problem of a nonlinear wave equation with damping and source terms. By the exact Gagliardo–Nirenberg inequality connected with the classic nonlinear elliptic equation, we establish new invariant sets of the problem. Then we get the exact conditions of blow-up and global existence.     

Rigorous Asymptotic Expansions for Critical Wave Speeds in a Family of Scalar Reaction-Diffusion Equations

We investigate traveling wave solutions in a family of reaction-diffusion equations which includes the Fisher–Kolmogorov–Petrowskii–Piscounov (FKPP) equation with quadratic nonlinearity and a bistable equation with degenerate cubic nonlinearity. It is known that, for each equation in this family, there is a critical wave speed which separates waves of exponential decay from those of algebraic decay at one of the end states. We derive rigorous asymptotic expansions for these critical speeds by perturbing off the classical FKPP and bistable cases. Our approach uses geometric singular perturbation theory and the blow-up technique, as well as a variant of the Melnikov method, and confirms the results previously obtained through asymptotic analysis in [J.H. Merkin and D.J. Needham, (1993). J. Appl. Math. Phys. (ZAMP) A, vol. 44, No. 4, 707–721] and [T.P. Witelski, K. Ono, and T.J. Kaper, (2001). Appl. Math. Lett., vol. 14, No. 1, 65–73].     

Blow-up of solution for a generalized Boussinesq equation

This paper studies the initial boundary value problem for a generalized Boussinesq equation and proves the existence and uniqueness of the local generalized solution of the problem by using the Galerkin method.Moreover,it gives the sufficient conditions of blow-up of the solution in finite time by using the concavity method.     

A computational investigation of the finite-time blow-up of the 3D incompressible Euler equations based on the Voigt regularization

We report the results of a computational investigation of two blow-up criteria for the 3D incompressible Euler equations. One criterion was proven in a previous work, and a related criterion is proved here. These criteria are based on an inviscid regularization of the Euler equations known as the 3D Euler–Voigt equations, which are known to be globally well-posed. Moreover, simulations of the 3D Euler–Voigt equations also require less resolution than simulations of the 3D Euler equations for fixed values of the regularization parameter \(\alpha >0\). Therefore, the new blow-up criteria allow one to gain information about possible singularity formation in the 3D Euler equations indirectly, namely by simulating the better-behaved 3D Euler–Voigt equations. The new criteria are only known to be sufficient criterion for blow-up. Therefore, to test the robustness of the inviscid-regularization approach, we also investigate analogous criteria for blow-up of the 1D Burgers equation, where blow-up is well known to occur.     

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 < x < π, with p > 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). Dedicated to Palo Brunovsky on the occasion of his birthday.     

Blow-up at the boundary for degenerate semilinear parabolic equations

This paper treats a superlinear parabolic equation, degenerate in the time derivative. It is shown that the solution may blow up in finite time. Moreover, it is proved that for a large class of initial data, blow-up occurs at the boundary of the domain when the nonlinearity is no worse than quadratic. Various estimates are obtained which determine the asymptotic behaviour near the blow-up. The mathematical analysis is then extended to equations with other degeneracies.     

Spectral Analysis of the Neumann–Poincaré Operator and Characterization of the Stress Concentration in Anti-Plane Elasticity

When holes or hard elastic inclusions are closely located, stress which is the gradient of the solution to the anti-plane elasticity equation can be arbitrarily large as the distance between two inclusions tends to zero. It is important to precisely characterize the blow-up of the gradient of such an equation. In this paper we show that the blow-up of the gradient can be characterized by a singular function defined by the single layer potential of an eigenfunction corresponding to the eigenvalue 1/2 of a Neumann–Poincaré type operator defined on the boundaries of the inclusions. By comparing the singular function with the one corresponding to two disks osculating to the inclusions, we quantitatively characterize the blow-up of the gradient in terms of explicit functions. In electrostatics, our results apply to the electric field, which is the gradient of the solution to the conductivity equation, in the case where perfectly conducting or insulating inclusions are closely located.     

Self-similar Blow-Up Profiles for a Reaction–Diffusion Equation with Critically Strong Weighted Reaction

Journal of Dynamics and Differential Equations - We classify the self-similar blow-up profiles for the following reaction–diffusion equation with critical strong weighted reaction and...     

Blow-up solutions of the generalized Boussinesq equation obtained by variational iteration method

This paper deals with obtaining explicit solutions of a generalized non-linear Boussinesq equation using He’s variational iteration method. Both finite and blow-up solutions can be obtained.     

Isolated Singularities of the 1D Complex Viscous Burgers Equation

The Cauchy problem for the 1D real-valued viscous Burgers equation u _t +uu _x  = u _xx is globally well posed (Hopf in Commun Pure Appl Math 3:201–230, 1950). For complex-valued solutions finite time blow-up is possible from smooth compactly supported initial data, see Poláčik and Šverák (J Reine Angew Math 616:205–217, 2008). It is also proved in Poláčik and Šverák (J Reine Angew Math 616:205–217, 2008) that the singularities for the complex-valued solutions are isolated if they are not present in the initial data. In this paper we study the singularities in more detail. In particular, we classify the possible blow-up rates and blow-up profiles. It turns out that all singularities are of type II and that the blow-up profiles are regular steady state solutions of the equation.