Nonlinear dissipative wave equations with space-time dependent potentials

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.