Global existence, asymptotic behavior, and finite time blow up of solutions for a class of generalized thermoelastic system with p -Laplacian

In this work, we examine the initial boundary value problem for the thermoelastic system with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p -Laplacian, subject to a novel nonlinearity condition within a bounded domain. First, we introduce the blow-up solution for the problem under consideration for the negative initial energy. By introducing a set of potential wells, we construct invariant sets of solutions for the thermoelastic system with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p -Laplacian. Under subcritical and critical initial energy scenarios, we derive the global existence and asymptotic behavior of weak solutions, as well as blow-up phenomena occurring within a finite time. Finally, we provide the lower and upper bounds of the blow-up time for the thermoelastic system with <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi>p</m:mi> </m:math> p -Laplacian.