Doctoral Dissertations

Date of Award

5-2009

Degree Type

Dissertation

Degree Name

Doctor of Philosophy

Major

Mathematics

Major Professor

Grozdena Todorova

Abstract

We study the long time behavior of solutions of the wave equations with absorption abs (u(t, x))[superscript p]⁻¹u(t, x) and variable damping a(t, x)u[subscript t](t, x), where p belongs to (1, n + 2/n - 2) and a(t, x) ~ a₀(1 + abs(x))⁻[superscript alpha](1 + t)⁻[superscript beta] for large abs x and t, a₀ > 0, for alpha belongs to (-infinity, 1), beta belongs to (-1, 1). We established decay estimates for the energy, L² and L[superscript p]⁺¹ norm of the solutions. 1. For alpha belongs to [0, 1), beta belongs to (-1, 1) and alpha + beta belongs to (0, 1), three different regimes of decay of solutions were found depending on the exponent of the absorption term, p₁(n, alpha, beta) := 1 + 4(beta + 1)/(2(n - alpha)(beta + 1) - beta(2 - alpha)) is a critical exponent in the following sense.For the supercritical region, namely p belongs to (p₁(n, alpha, beta), (n + 2)/(n - 2), the decay of solutions of the nonlinear equation coincides with the decay of the corresponding linear problem. For the subcritical region p belongs to (1,p₁(n, alpha, beta)) the decay is much faster. Moreover, the subcritical region is divided into two subregions with completely different decay rates by another critical exponent p₂ := 1 + (2alpha)/(n - alpha). If p belongs to (1, p₂(n, alpha, beta)) the decay of solutions becomes independent of alpha and beta. 2. For alpha belongs to (-infinity, 0) and beta belongs to (-1, 1). Two different regimes of decay of solutions were found depending on the exponent of the absorption term. p₁(n, alpha, beta) := 1 + 4(1 - beta)/(2(n + alpha)(1 - beta) + beta(2 + alpha)) is a critical exponent in the following sense.For the supercritical region, namely p belongs to (p₁(n, alpha, beta), (n + 2)/(n - 2)), the decay of solutions of the nonlinear equation coincides with the decay of the corresponding linear problem. For the subcritical region p belongs to (1, p₁(n, alpha, beta) the decay is much faster. We study also the long time behavior of solutions of the wave equations with focusing - abs (u(t, x))[superscript p]⁻¹u(t, x) and variable damping a(t, x)u[subscript t](t, x), where p belongs to (1, n + 2/n - 2) and a(t, x) ~ a₀(1 + absx)⁻[superscript alpha](1 + t)⁻[superscript beta] for large abs x and t, a₀ > 0, for alpha belongs to (0, 1), beta belongs to (-1, 1). A sharp critical exponents results were found depending on the exponent of the focusing term, for supercritical region, namely; p belongs to (p(n, alpha, beta) := 1 + 4(beta + 1)/2(n - alpha)(beta + 1) - beta(2 - alpha), n - 2/n + 2) the solutions are global for all small data.We also established decay estimates for the energy, L² and L[superscript p]⁺¹ norm of the solutions.

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