Solvability of the Nonlocal Initial Value Problem and Application to Design of Controller for Heat-equation with Delay

2019 ◽  
Vol 52 (2) ◽  
pp. 127-159 ◽  
Author(s):  
Xiao Pei Liu and Gen Qi Xu
Keyword(s):  
Heat Equation ◽  
Initial Value Problem ◽  
Initial Value ◽  
Equation With Delay

Dual extremum principles for the heat equation

1977 ◽  
Vol 77 (3-4) ◽  
pp. 273-292 ◽  
Author(s):  
W. D. Collins
Keyword(s):  
Hilbert Space ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Initial Value Problem ◽  
General Class ◽  
Initial Value Problems ◽  
Linear Operators ◽  
Point Boundary ◽  
Initial Value ◽  
Quantities Of Interest

SynopsisDual extremum principles characterising the solution of initial value problems for the heat equation are obtained by imbedding the problem in a two-point boundary-value problem for a system in which the original equation is coupled with its adjoint. Bounds on quantities of interest in the original initial value problem are obtained. Such principles are examples of ones which can be obtained for a general class of linear operators on a Hilbert space.

scholarly journals Remarks on the critical nonlinear high-order heat equation

2019 ◽  
Vol 26 (1/2) ◽  
pp. 127-152
Author(s):  
Tarek Saanouni
Keyword(s):  
Global Existence ◽  
Heat Equation ◽  
Initial Value Problem ◽  
Existence Of Solutions ◽  
High Order ◽  
Well Posedness ◽  
Potential Well ◽  
Initial Value ◽  
Global Existence Of Solutions ◽  
Nonlinear High Order

The initial value problem for a semi-linear high-order heat equation is investigated. In the focusing case, global well-posedness and exponential decay are obtained. In the focusing sign, global and non global existence of solutions are discussed via the potential well method.

scholarly journals A Unified Analytical Approach to Fixed and Moving Boundary Problems for the Heat Equation

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 749
Author(s):  
Marianito R. Rodrigo ◽  
Ngamta Thamwattana
Keyword(s):  
Heat Equation ◽  
Initial Value Problem ◽  
Moving Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Moving Boundary Problems ◽  
Initial Value ◽  
Boundary Problems ◽  
Fixed Boundary Problems ◽  
Initial Boundary Value

Fixed and moving boundary problems for the one-dimensional heat equation are considered. A unified approach to solving such problems is proposed by embedding a given initial-boundary value problem into an appropriate initial value problem on the real line with arbitrary but given functions, whose solution is known. These arbitrary functions are determined by imposing that the solution of the initial value problem satisfies the given boundary conditions. Exact analytical solutions of some moving boundary problems that have not been previously obtained are provided. Moreover, examples of fixed boundary problems over semi-infinite and bounded intervals are given, thus providing an alternative approach to the usual methods of solution.

Asymptotics for a class of heat equations with inhomogeneous nonlinearity

Analysis ◽  
2018 ◽  
Vol 38 (1) ◽  
pp. 21-36
Author(s):  
Tarek Saanouni ◽  
Radhia Ghanmi
Keyword(s):  
Heat Equation ◽  
Initial Value Problem ◽  
Energy Space ◽  
Nonlinear Heat Equation ◽  
Heat Equations ◽  
Well Posedness ◽  
Initial Value ◽  
Power Nonlinearity

AbstractThe initial value problem for an inhomogeneous nonlinear heat equation with a pure power nonlinearity is investigated. In the radial energy space, global and non-global well-posedness are discussed.

scholarly journals The heat semigroup and equation related to a Bessel-type operators and the canonical Fourier Bessel transform

2021 ◽  
Author(s):  
Ghazouani Sami ◽  
Sahbani Jihed
Keyword(s):  
Heat Equation ◽  
Initial Value Problem ◽  
Translation Operator ◽  
Direct Application ◽  
Heat Semigroup ◽  
Convolution Product ◽  
Initial Value ◽  
Bessel Transform

In this paper we study a translation operator associated with the canonical Fourier Bessel transform $\mathcal{F}_{\nu}^{\mathbf{m}}.$ We then use it to derive a convolution product and study some of its important properties. As a direct application, we introduce the heat semigroup generated by the Bessel-type operators $$\Delta_{\nu}^{\mathbf{m}^{-1}}=\frac{d^{2}}{dx^{2}}+\left( \frac{2\nu +1}{x}+2i \frac{a}{b} x\right) \frac{d}{dx}-\left( \frac{a^{2}}{b^{2}}x^{2}-2i\left( \nu +1\right) \frac{a}{b}\right) $$ and use it to solve the initial value problem for the heat equation governed by $\Delta_{\nu}^{\mathbf{m}^{-1}}.$

The semilinear heat equation with a Heaviside source term

1992 ◽  
Vol 3 (4) ◽  
pp. 367-379 ◽  
Author(s):  
Roberto Gianni ◽  
Josephus Hulshof
Keyword(s):  
Boundary Conditions ◽  
Heat Equation ◽  
Initial Value Problem ◽  
Source Term ◽  
Dirichlet Boundary ◽  
Dirichlet Boundary Conditions ◽  
Initial Value ◽  
Uniqueness Of Solutions ◽  
Bounded Interval ◽  
Semilinear Heat Equation

We consider the initial value problem for the equation ut = uxx + H(u), where H is the Heaviside graph, on a bounded interval with Dirichlet boundary conditions, and discuss existence, regularity and uniqueness of solutions and interfaces.

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