Well posedness and regularity for heat equation with the initial condition in weighted Orlicz–Slobodetskii space subordinated to Orlicz space like $$\lambda (\mathrm{log} \lambda )^\alpha $$ λ ( log λ ) α and the logarithmic weight

This paper is devoted to the numerical analysis of weak Galerkin mixed finite element method (WGMFEM) for solving a heat equation with random initial condition. To set up the finite element spaces, we choose piecewise continuous polynomial functions of degree j+1 with j≥0 for the primary variables and piecewise discontinuous vector-valued polynomial functions of degree j for the flux ones. We further establish the stability analysis of both semidiscrete and fully discrete WGMFE schemes. In addition, we prove the optimal order convergence estimates in L2 norm for scalar solutions and triple-bar norm for vector solutions and statistical variance-type convergence estimates. Ultimately, we provide a few numerical experiments to illustrate the efficiency of the proposed schemes and theoretical analysis.

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ASYMPTOTIC BEHAVIOR FOR TWO REGULARIZATIONS OF THE CAUCHY PROBLEM FOR THE BACKWARD HEAT EQUATION

Mathematical Models and Methods in Applied Sciences ◽

10.1142/s0218202598000093 ◽

1998 ◽

Vol 08 (01) ◽

pp. 187-202 ◽

Cited By ~ 30

Author(s):

K. A. AMES ◽

L. E. PAYNE

Keyword(s):

Asymptotic Behavior ◽

Heat Equation ◽

Singularly Perturbed ◽

Infinite Cylinder ◽

Initial Condition ◽

Damped Wave Equation ◽

Spatial Decay ◽

Initial Value ◽

The Cauchy Problem ◽

Negative Damping

One method of regularizing the initial value problem for the backward heat equation involves replacing the equation by a singularly perturbed hyperbolic equation which is equivalent to a damped wave equation with negative damping. Another regularization of this problem is obtained by perturbing the initial condition rather than the differential equation. For both of these problems, we investigate the asymptotic behavior of the solutions as the distance from the finite end of a semi-infinite cylinder tends to infinity and thus establish spatial decay results of the Saint-Venant type.

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Global Well-Posedness and Finite-Time Blow-Up of Some Heat-Type Equations

Proceedings of the Edinburgh Mathematical Society ◽

10.1017/s0013091516000213 ◽

2016 ◽

Vol 60 (2) ◽

pp. 481-497 ◽

Cited By ~ 3

Author(s):

Tarek Saanouni

Keyword(s):

Heat Equation ◽

Finite Time ◽

Blow Up ◽

Type Equation ◽

High Order ◽

Energy Space ◽

Well Posedness ◽

Exponential Nonlinearity

AbstractWe study two different heat-type equations. First, global well-posedness in the energy space of some high-order semilinear heat-type equation with exponential nonlinearity is obtained for even space dimensions. Second, a finite-time blow-up result for the critical monomial focusing heat equation with the p-Laplacian is proved.

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Bayesian Recovery of the Initial Condition for the Heat Equation

Communication in Statistics- Theory and Methods ◽

10.1080/03610926.2012.681417 ◽

2013 ◽

Vol 42 (7) ◽

pp. 1294-1313 ◽

Cited By ~ 18

Author(s):

B. T. Knapik ◽

A. W. van der Vaart ◽

J. H. van Zanten

Keyword(s):

Heat Equation ◽

Initial Condition

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On the Operator ⨁BkRelated to Bessel Heat Equation

Mathematical Problems in Engineering ◽

10.1155/2010/497676 ◽

2010 ◽

Vol 2010 ◽

pp. 1-12

Author(s):

Wanchak Satsanit

Keyword(s):

Heat Equation ◽

Heat Kernel ◽

Positive Constant ◽

Nonnegative Integer ◽

Unknown Function ◽

Generalized Function ◽

Initial Condition

We study the equation(∂/∂t)u(x,t)=c2⊕Bku(x,t)with the initial conditionu(x,0)=f(x)forx∈Rn+.The operator⊕Bkis the operator iterated k-times and is defined by⊕Bk=((∑i=1pBxi)4-(∑j=p+1p+qBxi)4)k, wherep+q=nis the dimension of theRn+,Bxi=∂2/∂xi2+(2vi/xi)(∂/∂xi),2vi=2αi+1,αi>-1/2,i=1,2,3,…,n, andkis a nonnegative integer,u(x,t)is an unknown function for(x,t)=(x1,x2,…,xn,t)∈Rn+×(0,∞),f(x)is a given generalized function, andcis a positive constant. We obtain the solution of such equation, which is related to the spectrum and the kernel, which is so called Bessel heat kernel. Moreover, such Bessel heat kernel has interesting properties and also related to the kernel of an extension of the heat equation.

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Propagação do calor em uma barra homogênea com condições de fronteira e dependendo de um parâmetro real

Ciência e Natura ◽

10.5902/2179460x24870 ◽

1981 ◽

Vol 3 (3) ◽

pp. 01

Author(s):

Lilian M. Kieling Reis ◽

Vanilde Bisognin

Keyword(s):

Analytical Solution ◽

Boundary Conditions ◽

Heat Equation ◽

Initial Condition

In this work the permanent temperature, in one homogeneous bar with boundary conditions that depends of a real parameters, was determined. The problem to be resolved was find the analytical solution of the heat equation ut =α2 uxx with the initial condition u (x, 0) = 0, ≤ x ≤ L and the contours conditions u (0, t) = 0 and u (L, t) = sen t, t> 0.

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The heat equation on line with random right part from Orlicz space

Carpathian Mathematical Publications ◽

10.15330/cmp.6.1.134-148 ◽

2014 ◽

Vol 6 (1) ◽

pp. 134-148 ◽

Cited By ~ 1

Author(s):

A.I. Slyvka-Tylyshchak

Keyword(s):

Heat Equation ◽

Random Field ◽

Orlicz Space ◽

Field Sample ◽

On Line ◽

The Right

In this paper the heat equation with random right part is examined. In particular, we give conditions for existence with probability one of the solutions in the case when the right part is a random field, sample continuous with probability one from the Orlicz space. Estimation for the distribution of the supremum of solutions of such equations is found.

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A Remark on the Heat Equation and Minimal Morse Functions on Tori and Spheres

Ingeniería y Ciencia ◽

10.17230/ingciecia.9.17.1 ◽

2013 ◽

Vol 9 (17) ◽

pp. 11-20

Author(s):

Carlos Cadavid ◽

Juan Diego Vélez

Keyword(s):

Riemannian Manifold ◽

Heat Equation ◽

Critical Points ◽

Morse Function ◽

Initial Condition ◽

Morse Functions ◽

Flat Tori

Let (M, g)be a compact, connected riemannian manifold that is homogeneous, i.e. each pair of pointsp, q∈M have isometric neighborhoods. This paper is a first step towards an understanding of the extent to which it is true that for each “generic” initial condition f0, the solution to∂f /∂t= ∆gf, f (·,0) =f0is such that for sufficiently larget, f(·, t) is a minimal Morse function, i.e., a Morse function whose total number of critical points is the minimal possible on M. In this paper we show that this is true for flat tori and round spheres in all dimensions.

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