Global Solution and Blow-up for a Thermoelastic System of $p$-Laplacian Type with Logarithmic Source

Abstract

This manuscript deals with global solution, polynomial stability and blow-up behavior at a finite time for the nonlinear system \begin{align*} \left\{ \begin{array}{rcl} & u'' - \Delta_{p} u + \theta + \alpha u' = \left\vert u\right\vert ^{p-2}u\ln \left\vert u\right\vert \\ &\theta' - \Delta \theta = u' \end{array}% \right. \end{align*} where $\Delta_{p}$ is the nonlinear $p$-Laplacian operator, $ 2 \leq p < \infty$. Taking into account that the initial data is in a suitable stability set created from the Nehari manifold, the global solution is constructed by means of the Faedo-Galerkin approximations. Polynomial decay is proven for a subcritical level of initial energy. The blow-up behavior is shown on an instability set with negative energy values.

Derivation of Emergent Spacetime Metric, Gravitational Potential and Speed of Light in Superfluid Vacuum Theory

“…where b(x, t) is a real-valued coupling function, D and ρ 0 are positive constants, and V(x, t) is external potential, often called trap potential in the conventional BEC theory. While the parameter ρ 0 is just a scaling one, the coupling function b is a dynamical value related to quantum temperature, a more detailed discussion can be found in [13] (there were also extensive mathematical studies of the logarithmic nonlinear models and around, to mention just very recent literature [14][15][16][17][18][19][20][21][22][23][24]). The constant parameter D can be written as…”