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Estimates for the nonlinear viscoelastic damped wave equation on compact Lie groups
- Part of
-
- Journal:
- Proceedings of the Royal Society of Edinburgh. Section A: Mathematics / Volume 154 / Issue 3 / June 2024
- Published online by Cambridge University Press:
- 19 May 2023, pp. 810-829
- Print publication:
- June 2024
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- Article
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Let $G$
be a compact Lie group. In this article, we investigate the Cauchy problem for a nonlinear wave equation with the viscoelastic damping on $G$
. More precisely, we investigate some $L^2$
-estimates for the solution to the homogeneous nonlinear viscoelastic damped wave equation on $G$
utilizing the group Fourier transform on $G$
. We also prove that there is no improvement of any decay rate for the norm $\|u(t,\,\cdot )\|_{L^2(G)}$
by further assuming the $L^1(G)$
-regularity of initial data. Finally, using the noncommutative Fourier analysis on compact Lie groups, we prove a local in time existence result in the energy space $\mathcal {C}^1([0,\,T],\,H^1_{\mathcal {L}}(G)).$
A DIFFUSIVE LOGISTIC EQUATION WITH MEMORY IN BESSEL POTENTIAL SPACES
- Part of
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- Journal:
- Bulletin of the Australian Mathematical Society / Volume 92 / Issue 2 / October 2015
- Published online by Cambridge University Press:
- 16 June 2015, pp. 251-258
- Print publication:
- October 2015
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- Article
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