Blow-up Time Research Articles

Lifespan of solutions to a parabolic type Kirchhoff equation with time-dependent nonlinearity

<p style='text-indent:20px;'>In this paper, an initial boundary value problem for a parabolic type Kirchhoff equation with time-dependent nonlinearity is considered. A new blow-up criterion for nonnegative initial energy is given and upper and lower bounds for the blow-up time are also derived. These results partially generalize some recent ones obtained by Han and Li in [Threshold results for the existence of global and blow-up solutions to Kirchhoff equations with arbitrary initial energy, Computers and Mathematics with Applications, 75(2018), 3283-3297].</p>

Some Results on Blow-up Phenomenon for Nonlinear Porous Medium Equations with Weighted Source

This paper deals with the blow-up phenomena for a type of nonlinear porous medium equations with weighted source ut −4um = a(x)f(u) subject to Dirichlet (or Neumann) boundary conditions. Based on the auxiliary functions and differential-integral inequalities, the blow-up criterions which ensure that u cannot exist all time are given under two different assumptions, and the corresponding estimates on the upper bounds for blow-up time and blow-up rate are derived respectively. Moreover, we use three different methods to determine the lower bounds for blow-up time and blow-up rate estimates if blow-up does occurs.

Global Existence and Blow-Up for a Kirchhoff-Type Hyperbolic Problem with Logarithmic Nonlinearity

This paper deals with a Kirchhoff-type hyperbolic problem with logarithmic nonlinearity. Firstly, by employing the classical Galerkin method and potential well method, we show the existence of the global weak solution with subcritical initial energy and critical initial energy. Secondly, we obtain the finite time blow up results with subcritical initial energy and critical initial energy by using the concavity method. In addition, under some suitable conditions, we also estimate the upper and lower bounds of the blow-up time for the blow-up solution by using the differential inequality technique. Finally, we establish a new finite time blow up condition which is independent of the depth of the potential well and we obtain the upper bound of the blow-up time for the blow-up solution. Furthermore, we also prove the weak solution can blow up in finite time at arbitrary initial energy level by using this condition.

Blow-up phenomena for a reaction diffusion equation with special diffusion process

ABSTRACT This paper is concerned with the blow-up property of solutions to an initial boundary value problem for a reaction diffusion equation with special diffusion processes. It is shown, under certain conditions on the initial data, that the solutions to this problem blow up in finite time, by combining Hardy inequality, ‘moving’ potential well methods with some differential inequalities. Moreover, the upper and lower bounds for the blow-up time are also derived when blow-up occurs.

Bounds for below-up time in a nonlinear generalized heat equation

ABSTRACT This paper deals with the blow-up phenomena of the solutions to an evolution heat equation with nonlinearities of variable exponent type. For a certain solution with positive initial energy, an upper bound for blow-up time is determined if the solutions blow-up.

Numerical blow-up for nonlinear diffusion equation with neumann boundary conditions

This work is concerned with the study of the numerical approximation for the nonlinear diffusion equation \( (u^{m})_t= u_{xx}, \ 0<x<1, \ t>0 \), under Neumann boundary conditions \( u_x(0,t)=0, \ u_x(1,t)=u^{\alpha}(1,t), \ t>0 \). First, we obtain a semidiscrete scheme by the finite differences method and prove the convergence of its solution to the continuous one. Then, we establish the numerical blow-up and the convergence of the numerical blow-up time to the theoretical one when the mesh size goes to zero. Finally, we illustrate our analysis with some numerical experiments.

Asymptotic Properties for a Semilinear Edge-Degenerate Parabolic Equation

The author deals with a semi-linear edge-degenerate parabolic equation, and proves that the solution increases exponentially under the initial energy J(u0) ≤ d, where d is the mountain-pass level. Moreover, the author estimates the blow-up time and the blow-up rate for the solution under J(u0) < 0.

Lower bound of blow-up time to a fourth order parabolic equation modeling epitaxial thin film growth

In this paper, we considered a class of fourth order equation modeling epitaxial thin film growth. We establish a blow up result for the initial and boundary value problem. Furthermore, the lower bound of the blow up time and the blow up rate are derived.

Control of blowup and approximate optimality conditions for the wave equations

We study the blowup problems for semilinear wave equations with control function in the domain and on the boundary. Sufficient optimality conditions for controlled blowup time are derived in terms of new dynamic programming methodology. We define ε-optimal value function and we construct sufficient ε-optimal conditions for that function again in terms of new dynamic programming inequality.

Bounds for lifespan of solutions to strongly damped semilinear wave equations with logarithmic sources and arbitrary initial energy

<p style='text-indent:20px;'>The main aim of this paper is to deal with the upper and lower bounds for blow-up time of solutions to the following equation: <p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ u_{tt}-\Delta u-\Delta u_{t} = |u|^{p-2}u\log|u|, $\end{document} </tex-math></disp-formula> <p style='text-indent:20px;'>which has been studied in [<xref ref-type="bibr" rid="b5">5</xref>]. For high initial energy, it is well known that the classical potential well method is not effective. In order to overcome this difficulty, the authors apply the new energy estimate method to establish the lower bound of the <inline-formula><tex-math id="M1">\begin{document}$ L^{2}(\Omega) $\end{document}</tex-math></inline-formula> norm of the solution. Furthermore, the authors construct a new control functional and combine energy inequalities with the concavity argument to prove that the solution blows up in finite time for high initial energy. Meanwhile, an estimate of the upper bound of blow-up time is also obtained. Finally, a lower bound for blow-up time is obtained by introducing a new control functional. These results fill the gap of [<xref ref-type="bibr" rid="b5">5</xref>].

Blow-Up Solution of a Porous Medium Equation with Nonlocal Boundary Conditions

In this paper, we devote to studying the blow-up phenomena for a porous medium equation under nonlocal boundary conditions. Based on auxiliary functions and differential inequality technique, we derive the sufficient conditions to guarantee the existence of blow-up solutions under different measures and obtain an upper bound for blow-up time. Moreover, we demonstrate the lower bounds for blow-up time under some appropriate measures in R3 and in the higher-dimensional space Rnn≥3, respectively. At last, two examples are given to illustrate the applications of main results.

Blow-Up Solutions to 3D Euler are Hydrodynamically Unstable

We study the interaction between the stability, and the propagation of regularity, for solutions to the incompressible 3D Euler equation. It is still unknown whether a solution with smooth initial data can develop a singularity in finite time. This article explains why the prediction of such a blow-up, via direct numerical experiments, is so difficult. It is described how, in such a scenario, the solution becomes unstable as time approaches the blow-up time.

Numerical validation of blow-up solutions with quasi-homogeneous compactifications

We provide a numerical validation method of blow-up solutions for finite dimensional vector fields admitting asymptotic quasi-homogeneity at infinity. Our methodology is based on quasi-homogeneous compactifications containing quasi-parabolic-type and directional-type compactifications. Divergent solutions including blow-up solutions then correspond to global trajectories of associated vector fields with appropriate time-variable transformation tending to equilibria on invariant manifolds representing infinity. We combine standard methodology of rigorous numerical integration of differential equations with Lyapunov function validations around equilibria corresponding to divergent directions, which yields rigorous upper and lower bounds of blow-up time as well as rigorous profile enclosures of blow-up solutions.

Lower Bound for the Blow-Up Time for the Nonlinear Reaction-Diffusion System in High Dimensions

In this paper, we study the blow-up phenomenon for a nonlinear reaction-diffusion system with time-dependent coefficients under nonlinear boundary conditions. Using the technique of a first-order differential inequality and the Sobolev inequalities, we can get the energy expression which satisfies the differential inequality. The lower bound for the blow-up time could be obtained if blow-up does really occur in high dimensions.

Bounds for the Blow-Up Time on the Pseudo-Parabolic Equation with Nonlocal Term

Bounds for the Blow-Up Time on the Pseudo-Parabolic Equation with Nonlocal Term

Blow-up instability in non-linear wave models with distributed parameters

We consider two model non-linear equations describing electric oscillations in systems with distributed parameters on the basis of diodes with non-linear characteristics. We obtain equivalent integral equations for classical solutions of the Cauchy problem and the first and second initial-boundary value problems for the original equations in the half-space . Using the contraction mapping principle, we prove the local-in-time solubility of these problems. For one of these equations, we use the Pokhozhaev method of non-linear capacity to deduce a priori bounds giving rise to finite-time blow-up results and obtain upper bounds for the blow-up time. For the other, we use a modification of Levine’s method to obtain sufficient conditions for blow-up in the case of sufficiently large initial data and give a lower bound for the order of growth of a functional with the meaning of energy. We also obtain an upper bound for the blow-up time.

Blow-Up Phenomena for a Fourth-Order Parabolic Equation with a General Nonlinearity

This paper is concerned with the blow-up property of solutions to an initial boundary value problem for a fourth-order parabolic equation with a general nonlinearity. It is shown, under certain conditions on the initial data, that the solutions to this problem blow up in finite time, using differential inequalities. Moreover, upper and lower bounds for the blow-up time are derived when blow-up occurs. This extends and generalizes results obtained by Philippin (Proc AMS. 2015;143(6):2507–13) and by Han (Nonlinear Anal RWA. 2018;43:451–66).

Lower bound for the blow-up time for a general nonlinear nonlocal porous medium equation under nonlinear boundary condition

In this paper, we study the blow-up phenomenon for a general nonlinear nonlocal porous medium equation in a bounded convex domain (varOmegain mathbb{R}^{n}, ngeq 3) with smooth boundary. Using the technique of a differential inequality and a Sobolev inequality, we derive the lower bound for the blow-up time under the nonlinear boundary condition if blow-up does really occur.

Critical and sup-critical initial energy finite time blowup phenomena for the fourth-order wave equations with nonlinear strain term

This paper is devoted to the blowup phenomena for a class of fourth-order wave equation with nonlinear strain term. First, for the model equation with strain structure, we prove the finite time blowup of solution for the critical initial energy, i.e., E(0)=d. Then, the finite time blowup and the upper bound of the blowup time are established for the sup-critical initial energy, i.e., E(0)>0.

Global and blow-up solutions for a nonlinear reaction diffusion equation with Robin boundary conditions

In the paper, we investigate global and blow-up solutions for a class of nonlinear reaction diffusion equations with Robin boundary conditions. By using auxiliary functions and a first-order differential inequality technique, we establish conditions on the data to prove the existence of global solution. Moreover, based on maximum principles, we obtain a criterion that guarantees the occurrence of the blow-up. When blow-up occurs, we discuss an upper bound and a lower bound on blow-up time. At last, we apply two examples to illustrate our main results.