Blow‐up results for a viscoelastic beam equation of <i>p</i>‐Laplacian type with strong damping and logarithmic source
Ducival Carvalho Pereira, Geraldo M. Araújo, Carlos A. Raposo, Victor R. Cabanillas
Abstract
In this paper, we investigate the existence, uniqueness, exponential decay, and blow‐up behavior of the viscoelastic beam equation involving the ‐Laplacian operator, strong damping, and a logarithmic source term, given by where is a bounded domain of and is a memory kernel. Using the Faedo–Galerkin approximation, we establish the existence and uniqueness result for the global solutions, taking into account that the initial data must belong to an appropriate stability set created from the Nehari manifold. The study of the exponential decay of our problem is based on Nakao's method. Finally, the blow‐up behavior on the instability set is proved.
Topics & Concepts
MathematicsUniquenessBounded functionMathematical analysisLogarithmExponential stabilityOperator (biology)Domain (mathematical analysis)Exponential decayLaplace operatorKernel (algebra)ViscoelasticitySemigroupPure mathematicsNonlinear systemPhysicsGeneThermodynamicsRepressorQuantum mechanicsNuclear physicsChemistryTranscription factorBiochemistryStability and Controllability of Differential EquationsAdvanced Mathematical Modeling in EngineeringAdvanced Mathematical Physics Problems