We study weak solutions of the Timoshenko equation in a bounded domain. We consider a nonlinear dissipation and a nonlinear source term. We obtain boundedness of the solutions as well as their asymptotic behavior. In particular, the source term does not produce a blowup, and the global attractor is the set of all equilibria.

In this paper, simultaneous control of source terms is considered in a vibrational string problem. In the considered problem, the terms to be controlled are the force and the initial velocity functions. We state the generalized (weak) solution about the considered problem. The existence and uniqueness of the solution for optimal control problem is investigated. The Frechet derivative of the functional and the Lipschitz continuity of the gradient are investigated. Minimizing sequence is obtained by the method of the projection of the gradient.

There is a general challenge in CFD research work either to implement density variation in incompressible algorithms or to solve constant-density fields using compressible algorithms. A desirable extension is the one which requires a minimum number of modifications and exhibits maximum performance. In this work, an easy two-step modification scheme is introduced in order to include density variation in a specific incompressible algorithm. The modifications result in new compressible source terms in the formulations which their behaviors are studied through the solution procedure.      

In this paper, we study the blow-up of solutions for hyperbolic equations involving the fractional Laplacian operator with damping and source terms.  We obtain the global existence results. Then, we observe the blow-up of solutions using the concavity method. Finally, we present some numerical simulation results.

In this paper, the focus is on investigating the asymptotic behavior of the solution for a system of parabolic equations with memory terms acting in both equations. This system has many applications in various scientific fields, including heat conduction in materials with memory effects and the study of biological systems exhibiting memory phenomena. The system of parabolic equations with a memory term provides a powerful framework for understanding and predicting the behavior of such complex systems, with emphasis on the role of the memory term in capturing the system's history-dependent behavior. Firstly, we assume that the relaxation functions $\mu_{2}\left( t\right) \leq\mu_{1}\left( t\right),\text{ for all} t\geq0$, and under certain conditions regarding the function p($ \cdot $) we prove that the solution with positive initial energy blows up in finite time. Finally, we present the theoretical results as numerical findings in the form of figures that illustrate and confirm the results by studying examples in two dimensions.