scholarly journals Schauder bases and the decay rate of the heat equation

2019 ◽  
Vol 19 (3) ◽  
pp. 717-728
Author(s):  
José Bonet ◽  
Wolfgang Lusky ◽  
Jari Taskinen
Keyword(s):  
Heat Equation ◽  
Decay Rate ◽  
Schauder Bases

A semilinear heat equation with singular initial data

1998 ◽  
Vol 128 (4) ◽  
pp. 745-758 ◽  
Author(s):  
Minkyu Kwak
Keyword(s):  
Initial Data ◽  
Heat Equation ◽  
Asymptotic Behaviour ◽  
Decay Rate ◽  
Existence And Uniqueness ◽  
Existence Of Solutions ◽  
Semilinear Heat Equation ◽  
Singular Initial Data ◽  
Self Similar ◽  
Image Position

We first prove existence and uniqueness of non-negative solutions of the equationin in the range 1 < p < 1 + 2/N, when initial data u(x, 0) = a|x|−2(p−1), x ≠ 0, for a > 0. It is proved that the maximal and minimal solutions are self-similar with the formwhere g = ga satisfiesAfter uniqueness is proved, the asymptotic behaviour of solutions ofis studied. In particular, we show thatThe case for a = 0 is also considered and a sharp decay rate of the above equation is derived. In the final, we reveal existence of solutions of the first and third equations above, which change sign.

SPATIAL DECAY FOR SOLUTIONS OF CAUCHY PROBLEMS FOR PERTURBED HEAT EQUATIONS

2001 ◽  
Vol 11 (05) ◽  
pp. 797-808 ◽  
Author(s):  
J. C. SONG
Keyword(s):  
Heat Equation ◽  
Decay Rate ◽  
Exponential Decay ◽  
Perturbation Parameter ◽  
Infinite Cylinder ◽  
Heat Equations ◽  
Cauchy Problems ◽  
Spatial Decay ◽  
Exponential Decay Rate ◽  
Decay Of Solutions

This paper investigates the decay of solutions of four different perturbations of the heat equation in a semi-infinite cylinder when it is heated at the finite end. In each case a bound for the decay of "energy" is obtained, with an explicit exponential decay rate which depends on the perturbation parameter.

Explicit decay rate for a degenerate hyperbolic-parabolic coupled system

2020 ◽  
Vol 26 ◽  
pp. 116
Author(s):  
Zhong-Jie Han ◽  
Gengsheng Wang ◽  
Jing Wang
Keyword(s):  
Diffusion Coefficient ◽  
Wave Equation ◽  
Heat Equation ◽  
Decay Rate ◽  
Semigroup Theory ◽  
Coupled System ◽  
Transmission Conditions ◽  
Polynomial Decay ◽  
Well Posedness ◽  
The Stability

This paper studies the stability of a 1-dim system which comprises a wave equation and a degenerate heat equation in two connected bounded intervals. The coupling between these two different components occurs at the interface with certain transmission conditions. We find an explicit polynomial decay rate for solutions of this system. This rate depends on the degree of the degeneration for the diffusion coefficient near the interface. Besides, the well-posedness of this degenerate coupled system is proved by the semigroup theory.

The Resolvent Operator

2017 ◽  
Author(s):  
Charles L. Epstein ◽  
Rafe Mazzeo
Keyword(s):  
Cauchy Problem ◽  
Heat Equation ◽  
Laplace Transform ◽  
Heat Kernel ◽  
Resolvent Operator ◽  
Contour Integration ◽  
Resolvent Kernel ◽  
The Laplace Transform ◽  
Elliptic Estimates ◽  
The Resolvent Operator

This chapter describes the construction of a resolvent operator using the Laplace transform of a parametrix for the heat kernel and a perturbative argument. In the equation (μ‎-L) R(μ‎) f = f, R(μ‎) is a right inverse for (μ‎-L). In Hölder spaces, these are the natural elliptic estimates for generalized Kimura diffusions. The chapter first constructs the resolvent kernel using an induction over the maximal codimension of bP, and proves various estimates on it, along with corresponding estimates for the solution operator for the homogeneous Cauchy problem. It then considers holomorphic semi-groups and uses contour integration to construct the solution to the heat equation, concluding with a discussion of Kimura diffusions where all coefficients have the same leading homogeneity.

Conserved Quantities for the General Linear Heat Equation

2016 ◽  
pp. 4437-4439
Author(s):  
Adil Jhangeer ◽  
Fahad Al-Mufadi
Keyword(s):  
Evolution Equation ◽  
Heat Equation ◽  
Conservation Laws ◽  
Exact Solutions ◽  
Lagrangian Approach ◽  
General Linear ◽  
Conserved Quantities ◽  
Linear Heat ◽  
The Family ◽  
Partial Lagrangian

In this paper, conserved quantities are computed for a class of evolution equation by using the partial Noether approach [2]. The partial Lagrangian approach is applied to the considered equation, infinite many conservation laws are obtained depending on the coefficients of equation for each n. These results give potential systems for the family of considered equation, which are further helpful to compute the exact solutions.

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