This work is concerned with the obtainment of new Carleman estimates for linear parabolic equations, where the second-order differential operator brings a super strong degeneracy in a positive measure subset of the spatial domain. In order to prove our main result, the control domain must contain the set of degeneracies. As a well-known consequence, we achieve a null controllability result in the current context. It is worth remarking that this hypothesis under the control domain is essential, otherwise the problem is not null controllable, in general. Also, our current result cannot be directly obtained from that where the degenracy occurs at the boundary.

This paper investigates the existence of a global attractor and an exponential attractor for a Kirchhoff wave equation with structural damping and memory terms. First, we establish the global well-posedness of the solutions using the semigroup method. Next, we prove the existence of a bounded absorbing set and derive a quasi-stable inequality. Finally, we show that the global attractor possesses a finite fractal dimension.

In this paper, we consider the approximate controllability of semilinear differential inclusion featuring multivalued nonlinearities with nonconvex values and the nonlocal conditions described by a multivalued map. Initially, we explore the existence result by assuming the underlying state space to be a reflexive Banach space. Then, we discuss the approximate controllability results in super-reflexive state spaces based on two different growth conditions on the multivalued nonlinearity. Incorporating multivalued maps with nonconvex values adds complexity to the studied differential inclusions yet enhances their practical applicability, serving as the core motivation for this research endeavor. We conclude by presenting an illustrative example that satisfies the criteria outlined in this paper.

In this paper we continue our study of nonlocal reaction-diffusion problems of gradient flow type that we developed in a previous paper [5] by using the notion of CP-structured reaction functionals as introduced in [2]. We consider here problems with times delays and DCP-structured reaction functionals as introduced in [3]. We present an existence result, investigate stochastic homogenization, and provide an application to spatial population dynamics.

In this paper, we investigate the 3D magneto-micropolar Bénard equations with nonlinear damping and partial viscosity. By employing energy methods and properties in anisotropic Lebesgue spaces, we establish the existence and uniqueness of global strong solutions when $ \beta \geq 4 $. Our results not only extend the current theoretical framework but also provide new perspectives and methodologies for further research in the dynamics of magneto-micropolar fluids.

We investigate the scattering behavior of global energy solutions to the inhomogeneous defocusing generalized Hartree equation $ i\partial_t \varphi+\Delta \varphi = |x|^{-{\varrho}}|\varphi|^{\beta-2}\bigl(J_\gamma *|\cdot|^{-{\varrho}}|\varphi|^{\beta}\bigr)\varphi, \qquad {\varrho}>0,\ \beta>2. $ This work complements our earlier study (Acta Appl. Math., 177, no. 6 (2022)) on the radial case by removing the spherical symmetry assumption under the additional condition $ x\varphi_0\in L^2 $. The novelty is to establish the scattering in weighted Sobolev spaces within the mass-subcritical range$ \beta_0<\beta\leq \frac{\gamma+2-2{\varrho}}{d}+1, $where $ \beta_0 $ denotes the Strauss exponent. The analysis relies on decay properties of global solutions, nonlinear estimates in Strichartz norms, and conformal transformation techniques. This result provides a refined description of the asymptotic behavior of energy solutions, showing that the nonlinear interaction becomes negligible at large times and that the dynamics approach those of the associated linear Schrödinger equation. The complementary case $ {\varrho} < 0 $ will be addressed in future work.

We study the existence and numerical computation of traveling wave solutions for a family of nonlinear higher-order Boussinesq evolution systems with a Hamiltonian structure. This general Boussinesq evolution system includes a broad class of homogeneous and non-homogeneous nonlinearities. We establish the existence of traveling wave solutions using the variational structure of the system and the concentration-compactness principle by P.-L. Lions, even though the nonlinearity could be non-homogeneous. For the homogeneous case, the traveling wave equations of the Boussinesq system are approximated using a spectral approach based on a Fourier basis, along with an iterative method that includes appropriate stabilizing factors to ensure convergence. In the non-homogeneous case, we apply a collocation Fourier method supplemented by Newton's iteration. Additionally, we present numerical experiments that explore cases in which the wave velocity falls outside the theoretical range of existence.

In this paper, we discuss the eigenvalue regularity criterion of the MHD equations with fractional magnetic diffusion in the three-dimensional whole space. We show that the results established by Neustupa-Penel[8,9,10] and Miller[7] can be extended to MHD equations with fractional magnetic diffusion. The proof involves combining the evolution equation for the strain tensor with the delicate structure of the nonlinear terms. As a by-product, we also obtain some new regularity criterion for the generalized MHD equations

Gaussian models, traditionally employed to describe transport phenomena, often fail to capture anomalous processes, prompting the need for advanced mathematical frameworks. The continuous-time random walk (CTRW) approach, utilizing distributed order fractional (DOF) derivatives defined as $\mathbb{D}^\nu g(t) = {\int}_0^1 \partial^{\alpha}_t g(t)\nu(\alpha)d\alpha$, provides a robust alternative for modeling anomalous transport. This paper addresses the reconstruction of mixed DOF weight functions $\nu(\alpha) = \sum_{j = 1}^I p_j \delta(\alpha - \alpha_j) + \mu_\nu(\alpha), I \in \mathbb{N}$, combining discrete and continuous components, in fractional partial differential equations. We propose an algorithm to identify $\nu$ and the function $u_\nu$ satisfying $\mathbb{D}^\nu u_\nu = \frac{\partial^2 u_\nu}{\partial x^2}$, with boundary conditions $u_\nu(0,t) = g(t)$, $u_\nu(1,t) = h(t)$, initial condition $u_\nu(x,0) = 0$, and overdetermined condition $u_\nu(x_0,t) = \phi(t)$. We establish the uniqueness of $\nu$ and use the nonlinear Tikhonov regularization method and develop numerical schemes to recover it from noisy data $g_\varepsilon$, $h_\varepsilon$, and $\phi_\varepsilon$. Our method enhances the modeling of complex anomalous transport, such as ultraslow diffusion, with numerical results validating its effectiveness.

This paper deals with the following parabolic-parabolic-parabolic chemotaxis system with indirect signal production and nonlinear sensitivity:$ \begin{equation} \begin{cases} u_t = \nabla\cdot((u+1)^\alpha\nabla u)-\chi\nabla\cdot(uv^k\nabla v)+u(a-bu), & t >0, \; x\in\Omega, \\ v_t = \Delta v-v+w, & t >0, \; x\in\Omega, \\ w_t = \Delta w-w+u, & t >0, \; x\in\Omega, \\ \frac{\partial u}{\partial\nu} = \frac{\partial v}{\partial\nu} = \frac{\partial w}{\partial\nu} = 0, & t >0, \; x\in\partial\Omega, \\ u(0, x) = u_0(x), \; \; v(0, x) = v_0(x), \; \; w(0, x) = w_0(x), & x \in\Omega, \end{cases} \end{equation} $where $ \Omega\subset\mathbb{R}^2 $ is a bounded domain with smooth boundary, $ \alpha, k, a, b>0, $ and $ \chi\in \mathbb{R}. $ The key to controlling the singular chemotaxis term $ -\chi\nabla\cdot(uv^k\nabla v) $ for proving the global existence of classical solutions to (1) is to ensure the boundedness of $ \|v\|_{L^\infty} $. To this end, we prove the boundedness of $ \|v\|_{L^\infty} $ by constructing the energy functional $ \int_ \Omega u\ln u. $ Combining this with Gagliardo-Nirenberg inequality, the boundedness of $ \|u\|_{L^2} $ is obtained. With the help of these estimates, we prove system (1) has a global bounded classical solution. Finally, if $ \chi>0, $ and $ b $ is appropriately large, it is shown that the solution exponentially converges to the constant stationary solution $ (\frac{a}{b}, \frac{a}{b}, \frac{a}{b}) $ as $ t\rightarrow \infty. $