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Heat equation for Sturm-Liouville operator with singular propagation and potential

Michael Ruzhansky, Alibek Yeskermessuly

Abstract

This article considers the initial boundary value problem for the heat equation with the time-dependent Sturm-Liouville operator with singular potentials. To obtain a solution by the method of separation of variables, the problem is reduced to the problem of eigenvalues of the Sturm-Liouville operator. Further on, the solution to the initial boundary value problem is constructed in the form of a Fourier series expansion. A heterogeneous case is also considered. Finally, we establish the well-posedness of the equation in the case when the potential and initial data are distributions, also for singular time-dependent coefficients.

Heat equation for Sturm-Liouville operator with singular propagation and potential

Abstract

This article considers the initial boundary value problem for the heat equation with the time-dependent Sturm-Liouville operator with singular potentials. To obtain a solution by the method of separation of variables, the problem is reduced to the problem of eigenvalues of the Sturm-Liouville operator. Further on, the solution to the initial boundary value problem is constructed in the form of a Fourier series expansion. A heterogeneous case is also considered. Finally, we establish the well-posedness of the equation in the case when the potential and initial data are distributions, also for singular time-dependent coefficients.
Paper Structure (4 sections, 7 theorems, 162 equations)

This paper contains 4 sections, 7 theorems, 162 equations.

Table of Contents

  1. Introduction
  2. Homogeneous heat equation
  3. Non-homogeneous heat equation
  4. Very weak solutions

Key Result

Theorem 2.1

Assume that $q \in L^\infty(0,1)$, $a\in L^\infty[0,T]$, and $a(t)\geq a_0>0$ for all $t\in [0,T]$. For any $k\in \mathbb{R}$, if the initial condition satisfies $u_0 \in W^k_\mathcal{L}(0,1)$ then the heat equation (C.p1) with the initial/boundary problem (C.p2)-(C.p3) has a unique solution $u\in C where the constants in these inequalities are independent of $u_0$, $\nu$, $q$ and $a$.

Theorems & Definitions (19)

  • Theorem 2.1
  • proof
  • Corollary 2.2
  • proof
  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Definition 4.1
  • Definition 4.2
  • Remark 4.3
  • ...and 9 more