The current paper discusses the global existence and asymptotic behavior of solutions of the following new nonlocal problem$$ u_{tt}-M\left(\displaystyle \int_{\Omega}|\nabla u|^{2}\, dx\right)\triangle u + \delta u_{t}= |u|^{\rho-2}u \log|u|, \quad \text{in}\ \Omega \times ]0, \infty[,  $$where\begin{equation*}M(s)=\left\{\begin{array}{ll}{a-bs, }&{\text{for}\ s \in [0, \frac{a}{b}[, }\\{0, }&{\text{for}\ s \in [\frac{a}{b}, +\infty[. }\end{array}\right. \end{equation*}If the initial data are appropriately small, we derive existence of global strong solutions and the  exponential decay of the energy.

Identifying  the unknown source terms in diffusion models, including nonlocal ones, is an active research area with significant applications in engineering and scientific fields such as population dynamics, biology, and physics. This study examines an inverse problem focused on recovering a space-dependent source term in a degenerate diffusion model that includes a nonlocal space term, using final-time measured data. As a first step, the inverse problem is reformulated as an optimization one by considering its solution as the minimizer of a well-defined objective function. The existence of a unique solution to the associated direct problem is discussed in a functional framework based on suitable weighted Sobolev spaces. After that, we prove the existence of a minimizer by means of standard arguments, and establish a first-order necessary optimality condition. Using this last one, we obtain some results concerning the stability and local uniqueness property. For the numerical reconstruction of the missing source term, we designed an algorithm based on the Landweber iterative method and showed its effectiveness by providing several numerical tests.

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In this article, we explore the discontinuous Galerkin finite element method for two-parametric singularly perturbed convection-diffusion problems with a discontinuous source term. Due to the discontinuity in the source term, the problem typically shows a weak interior layer. Also, the presence of multiple perturbation parameters in the problem causes boundary layers on both sides of the boundary. In this work, we develop the nonsymmetric discontinuous Galerkin finite element method with interior penalties to handle the layer phenomenon. With the use of a typical Shishkin mesh, the domain is discretized, and a uniform error estimate is obtained. Numerical experiments are conducted to validate the theoretical conclusions.