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Heat and wave type equations with non-local operators, I. Compact Lie groups

Wagner A. A. de Moraes, Joel E. Restrepo, Michael Ruzhansky

Abstract

We prove existence, uniqueness and give the analytical solution of heat and wave type equations on a compact Lie group $G$ by using a non-local (in time) differential operator and a positive left invariant operator (maybe unbounded) acting on the group. For heat type equations, solutions are given in $L^q(G)$ for data in $L^p(G)$ with $1<p\leqslant 2\leqslant q<+\infty$. We also provide some asymptotic estimates (large-time behavior) for the solutions. Some examples are given. Also, for wave type equations, we give the solution on some suitable Sobolev spaces over $L^2(G)$. We complement our results, by studying a multi-term heat type equation as well.

Heat and wave type equations with non-local operators, I. Compact Lie groups

Abstract

We prove existence, uniqueness and give the analytical solution of heat and wave type equations on a compact Lie group by using a non-local (in time) differential operator and a positive left invariant operator (maybe unbounded) acting on the group. For heat type equations, solutions are given in for data in with . We also provide some asymptotic estimates (large-time behavior) for the solutions. Some examples are given. Also, for wave type equations, we give the solution on some suitable Sobolev spaces over . We complement our results, by studying a multi-term heat type equation as well.
Paper Structure (9 sections, 6 theorems, 95 equations)

This paper contains 9 sections, 6 theorems, 95 equations.

Table of Contents

  1. Introduction
  2. Preliminary Results
  3. Non-local (in time) differential operators
  4. Compact Lie groups
  5. Heat type equations with non-local differential operators
  6. Examples
  7. Wave type equations with non-local differential operators
  8. Multi-term heat type equations
  9. Acknowledgements

Key Result

Theorem 1

If $u_0\in L^p(G)$ for $1<p\leqslant 2$ and the condition (need) is satisfied then there exists a unique solution $u\in \mathcal{C}([0,+\infty); L^q(G))$ for $2\leqslant q<+\infty$ of the Cauchy problem (Heat-intro). In particular, if the condition (asymtotic-trace) holds then for any $1<p\leqslant with the constant $C_{\alpha,\lambda,p,q}$ independent of $u_0$ and $t>0.$

Theorems & Definitions (16)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Theorem 4
  • proof
  • Remark 5
  • Remark 6
  • Theorem 7
  • proof
  • Remark 8
  • ...and 6 more