scholarly journals Critical initial conditions for spatially-distributed thermal explosions

1992 ◽  
Vol 33 (3) ◽  
pp. 350-362 ◽  
Author(s):  
A. A. Lacey ◽  
G. C. Wake
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Lower Bounds ◽  
Blow Up ◽  
Initial Conditions ◽  
Minimal Solution ◽  
Upper And Lower Bounds ◽  
Spatially Distributed ◽  
Thermal Explosions ◽  
New Criteria

AbstractThe problem of finding critical initial data which separate conditions leading to blow-up from those which give solutions tending to the (stable) minimal solution is considered. New criteria for blow-up and global existence are found; these are equivalent to obtaining upper and lower bounds respectively for the set of critical initial data.

scholarly journals Global Existence and Blow-Up for the Classical Solutions of the Long-Short Wave Equations with Viscosity

2021 ◽  
Vol 2021 ◽  
pp. 1-16
Author(s):  
Jincheng Shi ◽  
Shengzhong Xiao
Keyword(s):  
Global Existence ◽  
Life Span ◽  
General Model ◽  
Wave Equations ◽  
Blow Up ◽  
Initial Conditions ◽  
Short Wave ◽  
Classical Solutions ◽  
Global Classical Solutions

We are concerned with the global existence of classical solutions for a general model of viscosity long-short wave equations. Under suitable initial conditions, the existence of the global classical solutions for the viscosity long-short wave equations is proved. If it does not exist globally, the life span which is the largest time where the solutions exist is also obtained.

scholarly journals Newton-variational solution of the Frank–Kamenetskii thermal explosion problem

1989 ◽  
Vol 67 (3) ◽  
pp. 442-445 ◽  
Author(s):  
Avygdor Moise ◽  
Huw O. Pritchard
Keyword(s):  
Thermal Explosion ◽  
Newton Method ◽  
Lower Bounds ◽  
Linear Equations ◽  
Spherical Geometry ◽  
Variational Solution ◽  
Upper And Lower Bounds ◽  
Dimensionless Temperature ◽  
Temperature Excess ◽  
Thermal Explosions

The Newton method was shown by Vatsya to be suitable for solving the generalised elliptic problem, and we show that this approach can be used to treat the Frank–Kamenetskii model of a thermal explosion, by using a variational solution of the sequence of linear equations that are encountered in the Newton method. Convergence to the desired solution is rapid in the case of spherical geometry. The method produces converging upper and lower bounds to the critical value of the dimensionless heat production rate, δc, and lower bounds to the dimensionless temperature excess distribution function θ and its critical form θc. Keywords: thermal explosions, Frank–Kamenetskii model.

Gradient estimates for degenerate diffusion equations II

1982 ◽  
Vol 91 (3-4) ◽  
pp. 335-346 ◽  
Author(s):  
Nicholas D. Alikakos ◽  
Rouben Rostamian
Keyword(s):  
Initial Data ◽  
Lower Bounds ◽  
Space Time ◽  
Diffusion Equations ◽  
Upper And Lower Bounds ◽  
Gradient Estimates ◽  
Degenerate Diffusion ◽  
Lipschitz Continuous ◽  
Degenerate Diffusion Equations

SynopsisWe establish upper and lower bounds for various norms of solutions and their gradients for the equation ut = div (|∇u|m−1 ∇u) in ℝN in terms of the norms of the initial data. Based on the L∞ estimate of ∇u, we conclude that u(x, t) is Lipschitz continuous in space-time, for all t>0, whenever u(x,0) is in L1(ℝN).

Breaking symmetry in focusing nonlinear Klein-Gordon equations with potential

2018 ◽  
Vol 15 (04) ◽  
pp. 755-788
Author(s):  
Vladimir Georgiev ◽  
Sandra Lucente
Keyword(s):  
Initial Data ◽  
Global Existence ◽  
Ground State ◽  
Nonlinear Term ◽  
Blow Up ◽  
Critical Energy ◽  
Ground State Solution ◽  
Energy Space ◽  
Nirenberg Inequality ◽  
Klein Gordon

We study the dynamics for the focusing nonlinear Klein–Gordon equation, [Formula: see text] with positive radial potential [Formula: see text] and initial data in the energy space. Under suitable assumption on the potential, we establish the existence and uniqueness of the ground state solution. This enables us to define a threshold size for the initial data that separates global existence and blow-up. An appropriate Gagliardo–Nirenberg inequality gives a critical exponent depending on [Formula: see text]. For subcritical exponent and subcritical energy global existence vs blow-up conditions are determined by a comparison between the nonlinear term of the energy solution and the nonlinear term of the ground state energy. For subcritical exponents and critical energy some solutions blow-up, other solutions exist for all time due to the decomposition of the energy space of the initial data into two complementary domains.

TRIDIAGONAL SOLVERS WITH MULTIPLE RIGHT HAND SIDES ON k-DIMENSIONAL MESH AND TORUS INTERCONNECTION NETWORKS

2003 ◽  
Vol 13 (04) ◽  
pp. 659-672
Author(s):  
EUNICE E. SANTOS
Keyword(s):  
Initial Data ◽  
Linear Systems ◽  
Lower Bounds ◽  
Execution Time ◽  
Interconnection Networks ◽  
Constant Factor ◽  
Upper And Lower Bounds ◽  
Cyclic Reduction ◽  
Right Hand ◽  
Small Constant

We consider the problem of designing optimal and efficient algorithms for solving tridiagonal linear systems with multiple right-hand side vectors on k-dimensional mesh and torus interconnection networks. We derive asymptotic upper and lower bounds for these solvers using odd-even cyclic reduction. We present various important bounds on execution time including general lower bounds which are independent of initial data assignment, and lower bounds based on classifying assignments via the proportion of initial data assigned amongst processors. Finally, different algorithms are designed in order to achieve running times that are within a small constant factor of the lower bounds provided.

scholarly journals Blow-up phenomena for the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Jinxing Liu ◽  
Xiongrui Wang ◽  
Jun Zhou ◽  
Huan Zhang
Keyword(s):  
Lower Bounds ◽  
Fourth Order ◽  
Boussinesq Equation ◽  
Blow Up ◽  
Sixth Order ◽  
Upper And Lower Bounds ◽  
Differential Inequalities ◽  
Nonlinear Source ◽  
Order Dispersion ◽  
Dispersion Term

<p style='text-indent:20px;'>This paper deals with the sixth-order Boussinesq equation with fourth-order dispersion term and nonlinear source. By using some ordinary differential inequalities, the conditions on finite time blow-up of solutions are given with suitable assumptions on initial values. Moreover, the upper and lower bounds of the blow-up time are also investigated.</p>

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