In this work, we consider a nonlinear transmission problem in the bounded domain with a delay term in the first equation. Under conditions on the weight of the damping and the weight of the delay, we prove general stability estimates by introducing a suitable Lyapunov functional and using the properties of convex functions.

In the present paper, we considered a Korteweg-de Vries type system with weakly damping terms and initial data in the analytic Gevery spaces. The presence of tow functions $c_1(x),c_2(x)$, called damping coefficients, made the system more interesting from an application point of view due to their great importance in physics. To start, by using the fixed point theorem in Banach space, we investigated the local well-posedness. Additionally, by employing an approximate conservation law, we extended this to be global in time, ensuring that the radius of analyticity of solutions remained uniformly bounded below by a fixed positive number for all time.

This paper discussed the decay of a thermoelastic laminated beam subjected to nonlinear delay and nonlinear structural damping. We provided explicit and general energy decay rates of the solution by imposing suitable conditions on both weight delay and wave speeds. To achieve this, we leveraged the properties of convex functions and employed the multiplier technique as a specific approach to demonstrate our stability results.

The problem of existence, uniqueness, and stability in the Hyers-Ulam sense of solutions for random impulsive stochastic functional differential equations driven by Poisson jumps with finite delays is considered. Based on techniques combining the generalized Banach fixed-point theorem and the expansion of the Perov-type fixed-point theorem, two significant quantitative and qualitative results are analyzed, and then an example is presented to illustrate our results.

In the present paper, we studied a hyperbolic $p(x)$-biharmonic problem with known no-flux values on the smooth part of the boundary in variable exponent Sobolev spaces. The global existence of a weak solution, using the Galerkin method, was proved. The blow-up of the solution with negative initial functional energy in finite time was discussed using the concavity method. The interaction between the variable exponent in the main equation and the boundary condition is crucial in determining the presence/absence of solutions.

In this article, a Cauchy problem for a coupled system of the generalized Korteweg-de Vries equations (gKdV) is considered. In the periodic case, it is shown that the system is locally well-posed in a large class of analytic functions and conditions for which weak solutions extend holomorphically in a symmetric strip of the complex plane around the $x$-axis at large times. In addition, the uniform analyticity radius of the solution does not change as time progresses. Also, information about the regularity of the solution in the time variable is obtained.