Blow up and Decay for a Class of $p$-Laplacian Hyperbolic Equation with Logarithmic Nonlinearity

Abstract

In this paper, we study an initial boundary value problem for a p-Laplacian hyperbolic equation with logarithmic nonlinearity. By combining the modified potential well method with the Galerkin method, the existence of the global weak solution is studied, and the polynomial and exponential decay estimation under certain conditions are also given. Moreover, by using the concavity method and other techniques, we obtain the blow up results at finite time.

Concavity Method: A concise survey

“…Apart from parabolic and pseudo-parabolic equations, the method plays a significant role in studying unbounded solutions of hyperbolic equations and systems. Some of the latest works are as follows [6,7,44,50,51]. Hence the Concavity method is a simple and powerful tool to use in the blow up studies of solutions to differential equation problems.…”

Blow‐up results for a viscoelastic beam equation of p‐Laplacian type with strong damping and logarithmic source

“…Regarding the analysis of this type of model, we mention the work of Pişkin et al, 25 who obtained global existence and decay estimates for the solutions of the following equation: $$$$ {u}_{tt}-{\Delta}_pu+{\left|u\right|}^{p-2}u+{u}_t={\left|u\right|}^{p-2}u\ln \left|u\right|. $$$$ Chu et al 26 studied the equation $$$$ {u}_{tt}-{\Delta}_pu-\Delta {u}_t={\left|u\right|}^{q-2}u\ln \left|u\right|, $$$$ and combining the modified potential well method with the Galerkin method proved the existence of a weak global solution, obtaining polynomial and exponential decay estimates under certain conditions. In addition, using the concavity method and other techniques, he obtained a blow‐up in finite time result.…”

Solutions for Some Specific Mathematical Physics Problems Issued from Modeling Real Phenomena: Part 2

“…Global weak solutions for evolution problems with logarithmic nonlinearities involving the p-pseudo-Laplacian are presented in [7][8][9][10], and those with p-Laplacian are studied in [11]. The existence of periodic solution [12], radial symmetry [13], symmetry [14], or principal eigenvalues [15] for fractional p-Laplacian, minimizers [16], and Picone type identities for p-Laplacian [17] or p-pseudo-Laplacian [18], regularity, and multiplicity results [19] represent interesting and topical issues that have important implications.…”