Global in time existence of Sobolev solutions to semi-linear damped sigma-evolution equations in L^q scales

The main goal of this thesis is to prove the global (in time) existence of small data Sobolev solutions to semi-linear damped σ-evolution equations from suitable function spaces basing on L^q spaces by mixing additional L^m regularity for the data on the basis of L^q-L^q estimates for solutions, with q∈(1,∞) and m∈[1,q), to the corresponding linear models. To establish desired results, we would like to apply the theory of modified Bessel functions, Faà di Bruno's formula and Mikhlin-Hörmander multiplier theorem in the treatment of linear problems. In addition, some of modern tools from Harmonic Analysis play a fundamental role to investigate results for the global existence of small data Sobolev solutions to semi-linear problems. Finally, the application of a modified test function method is to devote to the proof of blow-up results for semi-linear damped σ-evolution models, where σ≥1 and δ∈[0,σ) are assumed to be any fractional numbers

On the Cauchy problem for a weakly coupled system of semi-linear σ\sigma-evolution equations with double dissipation

In this paper, we would like to consider the Cauchy problem for a multi-component weakly coupled system of semi-linear $\sigma$-evolution equations with double dissipation for any $\sigma\ge 1$. The first main purpose is to obtain the global (in time) existence of small data solutions in the supercritical condition by assuming additional $L^1$ regularity for the initial data and using multi-loss of decay wisely. For the second main one, we are interested in establishing the blow-up results together with sharp estimates for lifespan of solutions in the subcritical case. The proof is based on a contradiction argument with the help of modified test functions to derive the upper bound estimates. Finally, we succeed in catching the lower bound estimate by constructing Sobolev spaces with the time-dependent weighted functions of polynomial type in their corresponding norms.Comment: 19 page