Solutions Of Nonlinear Heat Equation Research Articles

Global solution of nonlinear Heat equation with solutions in a Hilbert Manifold

The objective of this paper is to deal with the deterministic problem consisting of non-linear heat equation of gradient type. The evolution equation emerges as projecting the Laplace operator with Dirichlet boundary conditions and polynomial nonlinearity of degree 2n−1, onto the tangent space of a sphere M in a Hilbert space ℋ. We are going to deal with questions of the existence and the uniqueness of a global solution, and the invariance of the manifold, M i.e. if the suitable initial data lives on M then all trajectories of solutions also belong to M. Finally, we will also show the global solution generates a gradient flow.

The approximate solution of nonlinear heat equation

AbstractNowadays there are many approximate methods for thermal conductivity calculation, that lead to satisfactory outcomes in engineering practice. With the new approach, the solution of the nonlinear thermal conductivity equation was investigated. In addition to this, we reviewed the approximate solution of the nonlinear equation of thermal conductivity with cubic nonlinearity. To solve the problems of mathematical analysis, differential and integral equations, and boundary value problems of mathematical physics, difference, and interpolation are applied. Thus, for thinking of the effectiveness and reasonableness of these approaches, it is crucial for their theoretical investigation. The solution to these questions was found for each class of equation and each of its methods in their way and was often represented as significant difficulty, and in many cases was an obstacle for the current time. A natural approach to solving this issue is the use of the ideas of functional analysis. The variational principle initially was considered as a variational approach for solving linear functional equations and finding eigenvalues of linear operators. As in any variational approach, the problem of solving an equation will be brought to finding the extremum of the certain function of a special type, given over the entire space. It was found that the approach is useful in a way of minimizing functions of the more general type.

Modulation theory for the flat blow-up solutions of nonlinear heat equation

Modulation theory for the flat blow-up solutions of nonlinear heat equation

Qualitative analysis of solutions for a Kirchhoff-type parabolic equation with multiple nonlinearities

In this work, the local and global existence of weak solutions by using the Faedo-Galerkin method, the finite time blow up of the weak solution with positive initial energy and the general decay of the solution are discussed. Finally, we consider the exponential growth of the solution with sufficient conditions. This work generalizes and improves earlier results in the literature, see [L.X. Truong and N. Van Y, On a class of nonlinear heat equations with viscoelastic term, Comput. Math. Appl., 2016] and [L.X. Truong and N. Van Y, Exponential growth with ${L}^{p}$-norm of solutions for nonlinear heat equations with viscoelastic term, Appl. Math. Comput., 2016].

Universal bound for global solution of nonlinear heat equation

The aim of this paper is to study some properties of positive solutions to the nonlinear diffusion equation $$\begin{aligned} \frac{\partial u(x,t)}{\partial t} = \Delta _p u(x,t) + c(x)f(u(x,t)), \;\; (x,t) \in \Omega \times (0,\infty ). \end{aligned}$$ Assuming that f is of a bistable type with stable constant steady states 0 and $$c_0 >0$$ , we show, that there exist a universal, a priori upper bound for all positive solutions of the previous equation. Moreover, we prove the convergence of these solutions to the constant $$c_0$$ as t tends to $$+\,\infty $$ . Some examples where our results can be applied are provided.

Blow-up of Solutions of Nonlinear Heat Equation with Hypoelliptic Operators on Graded Lie Groups

In this paper we show blow-up of solutions of the nonlinear heat equation with the Rockland operators on the graded Lie groups. In addition, we give the necessary conditions for the existence of local or global solutions of the nonlinear heat equation with hypoelliptic operators on Heisenberg groups.

Blowup versus global in time existence of solutions for nonlinear heat equations

This note is devoted to a simple proof of blowup of solutions for a nonlinear heat equation. The criterion for a blowup is expressed in terms of a Morrey space norm and is in a sense complementary to conditions guaranteeing the global in time existence of solutions. The method goes back to H. Fujita and extends to other nonlinear parabolic equations.

Exponential growth with Lp-norm of solutions for nonlinear heat equations with viscoelastic term

In this work, we study an initial boundary value problem related to the nonlinear viscoelastic equation(1+a|u|q−2)ut−Δu+∫0tg(t−s)Δu(s)ds=b|u|p−2u.We show the exponential growth of solutions with Lp-norm for negative or positive initial energy.

An efficient numerical technique for the solution of nonlinear heat equation via spectral method

Nonlinear wave equations are more difficult to study mathematically, and no general analytical method exists for their solution. It is found that the Exponential Time Differencing (ETD) scheme requires the steps to achieve a given accuracy, offers a speedy method in calculation time, and has exceptional stability properties in solving a nonlinear equation. This article solves the diagonal example of nonlinear heat equation via the exponential time difference Runge-Kutta 4 methods (ETDRK4). Implementation of the method is proposed by short Matlab programs.

Solution of the Nonlinear Heat Equation for Circular-Foil Heat Flux Gauges

A series solution of the steady nonlinear heat equation for a circular-thin-foil heat flux gauge subjected to a constant-radiation heatflux is presented. The dimensionless formof the heat equation involves a dimensionless gauge parameter and a dimensionless irradiation parameter. A linear solution is extracted from the series solution, which yields a simple-yet-accurate formula that is useful for calibration of thin-foil heat flux gauges. The dimensionless temperature at the center of the foil is determined from the linear solution as a function of the preceding two parameters. The dependence of the irradiation parameter on the dimensionless temperature at the center of the circular foil is presented graphically for awide range of the parameters. The values of the dimensionless temperature at the center of the foil obtained with the linear solution are in good agreement with those obtained with a numerical solution of the heat equation usingRunge–Kutta integration. Typical temperature profiles also showgoodagreement with the numerical solution. To compensate for inherent gauge errors and assumptions made in the analysis, individual gauges are calibrated against standards. To curve fit the calibration data, it is suggested that a correction factor be introduced in the argument of the modified Bessel function that appears in the linear formula for the dimensionless irradiation parameter.

Adomian Decomposition Method for a Nonlinear Heat Equation with Temperature Dependent Thermal Properties

The solutions of nonlinear heat equation with temperature dependent diffusivity are investigated using the modified Adomian decomposition method. Analysis of the method and examples are given to show that the Adomian series solution gives an excellent approximation to the exact solution. This accuracy can be increased by increasing the number of terms in the series expansion. The Adomian solutions are presented in some situations of interest.

Functional Separable Solutions of Nonlinear Heat Equations in Non-Newtonian Fluids

We study the functional separation of variables to the nonlinear heat equation: ut = (A(x)D(u)uxn)x + B(x)Q(u), Ax ≠ 0. Such equation arises from non-Newtonian fluids. Its functional separation of variables is studied by using the group foliation method. A classification of the equation which admits the functional separable solutions is performed. As a consequence, some solutions to the resulting equations are obtained.

Study of exact solutions of nonlinear heat equations

One of the ways that energy transports in fluid is electron thermal conduction. The aim of Inertial Confinement Fusion (ICF) is to show that the thermal conductivity is strongly dependent on temperature and the equation of heat condition is a nonlinear equation. In this article, we analyze the exact solutions of the nonlinear equation of heat conduction problem with variable transfer coefficients which is the problem of ICF.

Blow-up in nonlinear heat equations

We study the blow-up of solutions of nonlinear heat equations in dimension 1. We show that for an open set of even initial data which are characterized roughly by having maxima at the origin, the solutions blow up in finite time and at a single point. We find the universal blow-up profile and remainder estimates. Our results extend previous results in several directions and our techniques differ from the techniques previously used for this problem. In particular, they do not rely on maximum principle.

Solitary wave and other solutions for nonlinear heat equations

Abstract A number of explicit solutions for the heat equation with a polynomial non-linearity and for the Fisher equation is presented. An extended class of non-linear heat equations admitting solitary wave solutions is described. The generalization of the Fisher equation is proposed whose solutions propagate with arbitrary ad hoc fixed velocity.

A sufficient condition for blowup solutions of nonlinear heat equations

The author discusses the initial-boundary value problem ( u i ) t = Δu i + f i ( u 1,…, u m ) with u i| ∂Ω=0 and u i ( x,0)= φ i ( x), i=1,…, m, in a bounded domain Ω⊂R n . Under suitable assumptions on f i , he proves that, if φ i ⩾(1+ ε 0) ψ i in D i⊂Ω , for some small ε 0>0, then the solutions blow up in a finite time, where ψ i is a positive solution of Δψ i + f i ( ψ 1,…, ψ m )⩾0, with ψ i | ∂D i =0 for i=1,…, m. If m=1, the initial value can be negative in a subset of Ω.

On Uniqueness and Stability of Self-Similar Blow-Up Solutions of Nonlinear Heat Equations: Evolution Approach

We present evolution arguments of studying uniqueness and asymptotic stability of blow-up self-similar solutions of second-order nonlinear parabolic equations from combustion and filtration theory. The analysis uses intersection comparison techniques based on the Sturm Theorem on zero set for linear parabolic equations. We show that both uniqueness and stability of similarity ODE profiles are directly related to the asymptotic structure of their domain of attraction relative to the corresponding parabolic evolution.

Composite structure of global unbounded solutions of nonlinear heat equations with critical Sobolev exponents

We describe special asymptotic structures of solutions of the semilinear heat equation u t= Δu+u p in Ω×(0,T), u=0 on ∂Ω×(0,T), in the unit ball Ω={r=|x|<1}⊂ R N in dimensions N⩾3 with positive symmetric initial data u 0( r). It is known (J. Differential Equations 54 (1984) 97) that in the critical Sobolev case p= p s =( N+2)/( N−2) the Cauchy problem with specially chosen initial data u 0 admits global unbounded solutions (GUSs) u(·, t), which are uniformly bounded in L 1(Ω) but are not bounded in L ∞(Ω) for all t>0. In the radial geometry, we establish the following asymptotic behaviour of such solutions as t→∞: ln ||u(·,t)|| ∞∼π 2t/4 for N=3, ln||u(·,t)|| ∞∼2 t for N=4, and || u(·, t)|| ∞∼ γ 0 t ( N−2)/2( N−4) for N⩾5, where γ 0= γ 0( N)>0 is a constant independent of initial data. The large-time behaviour is not self-similar and is obtained by matching of inner and outer asymptotic expansions. The phenomenon of GUSs is shown to be a common feature of a number of quasilinear and fully nonlinear parabolic equations with scaling invariant operators in the critical Sobolev case.

On uniqueness for ODEs Arising in Blow-up Asymptotics for Nonlinear Heat Equations

Abstract We study uniqueness for nonlinear ordinary differential equations arising in constructing blow-up and extinction self-similar solutions of various reaction-diffusion- absorption equations. Such particular similarity solutions describe the asymptotic singular behaviour of wide classes of general solutions of nonlinear heat equations. We prove that under some monotonicity assumptions, such similarity profiles are unique.

O.D.E. type behavior of blow-up solutions of nonlinear heat equations

We show that all global (in time and in space) and bounded solutions of the vector-valued equation <p align="center"> $\frac{\partial w}{\partial s}=\Delta w-\frac{1}{2}y \cdot \nabla w -\frac{w}{p-1}+|w|^{p-1}w$ <p align="left" class="times"> (where $w : \mathbb R^N\times \mathbb R \to \mathbb R^M, p> 1$ and $(N - 2)p < N + 2$) are independent of space and completely explicit. We then derive from this various uniform estimates and a uniform localization property for blow-up solutions of $\partial_t u=\Delta u + |u|^{p-1}u$.