Some results for a weakly coupled system of semi-linear structurally damped σ-evolution equations

In this paper, we are interested in studying the Cauchy problem for a weakly coupled system of two semi-linear structurally damped <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>σ</m:mi> <m:mi>k</m:mi> </m:msub> </m:math> {σₖ} -evolution equations, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>σ</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {σₖ≥ 1} for <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mrow> </m:math> {k=1,2} . Our first purpose involves the proof of global (in time) existence of small data energy solutions by mixing additional <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msup> <m:mi>L</m:mi> <m:msub> <m:mi>m</m:mi> <m:mi>k</m:mi> </m:msub> </m:msup> </m:math> {L^{mₖ}} regularity for the data, where <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>m</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>∈</m:mo> <m:mrow> <m:mo stretchy="false">[</m:mo> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> </m:math> {mₖ∈[1,2)} . We want to point out that in some cases of choosing suitable parameters <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>m</m:mi> <m:mi>k</m:mi> </m:msub> </m:math> {mₖ} , with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mi>k</m:mi> <m:mo>=</m:mo> <m:mrow> <m:mn>1</m:mn> <m:mo>,</m:mo> <m:mn>2</m:mn> </m:mrow> </m:mrow> </m:math> {k=1,2} , the obtained lower bound of one exponent p or q related to power nonlinearities on the right-hand sides is really smaller than the critical exponent, the so-called modified Fujita exponent. The second aim of this paper is to discuss a blow-up result for Sobolev solutions with any different fractional values of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>σ</m:mi> <m:mi>k</m:mi> </m:msub> <m:mo>≥</m:mo> <m:mn>1</m:mn> </m:mrow> </m:math> {σₖ≥ 1} when <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:msub> <m:mi>m</m:mi> <m:mn>1</m:mn> </m:msub> <m:mo>=</m:mo> <m:msub> <m:mi>m</m:mi> <m:mn>2</m:mn> </m:msub> </m:mrow> </m:math> {m₁=m₂} .