Abstract In this paper we derive a nonlinear theory of Mindlin’s Form II gradient for thermoelastic materials where the heat propagation model proposed by Green and Naghdi, taking into account micro-inertia effects as well. The elastic behavior is assumed to be consistent with Mindlin’s Form II gradient elasticity theory, while the thermal behavior is based on type III entropy balance, where the second gradient of the thermal displacement is included in the set of independent constitutive variables. This leads to a fourth-order equations for the temperature. The equations of the linear theory are also obtained. Semigroup theory is then used to prove the well-posedness of the obtained problem. We show that, in general, the one-dimensional problem is not exponentially stable but that there exists polynomial stability with rates that depend on the micro-inertia parameter h and the regularity of the initial data. Using a resolvent criterion developed by Borichev and Tomilov, we prove that the polynomial decay rate is t −1/3 when h > 0, while it is t −1 when h = 0. By following a result due to Arendt-Batty, we show that the considered problem is strongly stable whenever the value of h .

Abstract We propose in this paper a class of generalized nonlinear mixed variational inequality problems on Banach spaces, which is composed of three elliptic generalized nonlinear mixed variational inequality problems. We demonstrate that, under appropriate assumptions, the ideas of the Kakutani-Ky Fan fixed point theorem and Minty techniques, the existence of the solution set for generalized nonlinear mixed variational inequality problem is nonempty, weakly compact, and unique. Furthermore, we provide a stability analysis by perturbing the duality mappings and addressing the optimal control problem for the solution set of the system of generalised nonlinear mixed variational inequality problems.

Abstract This study addresses the problem of indirect boundary observability for a spatial semi-discretization of weakly coupled two-dimensional wave equations. The discretization is performed using a finite difference method on a uniform mesh. More specifically, the analysis focuses on the uniformity of the observability inequality with respect to the discretization parameter as it tends to zero. It is well known that high-frequency numerical solutions emerge when wave equations are discretized via finite differences on uniform grids. Consequently, the uniform observability property fails to hold, as the observability constant typically becomes unbounded when the mesh size decreases. This lack of uniform observability poses a significant challenge in numerical control theory. To mitigate this issue, we employ a Fourier filtering technique to eliminate spurious high-frequency components. By adapting calculus techniques analogous to those used in the continuous setting, we establish a uniform observability inequality for the filtered semi-discrete system.

Abstract In this paper, we consider the non-isentropic compressible Euler equations with time-dependent damping − α ( t ) u in one space dimension <m:math xmlns:m="http://www.w3.org/1998/Math/MathML" display="block" overflow="scroll"> <m:mfenced close="" open="{"> <m:mrow> <m:mtable class="cases"> <m:mtr> <m:mtd columnalign="left"> <m:msub> <m:mrow> <m:mi>v</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mo>−</m:mo> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:msub> <m:mrow> <m:mi>u</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mo>+</m:mo> <m:mi>p</m:mi> <m:msub> <m:mrow> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>v</m:mi> <m:mo>,</m:mo> <m:mi>s</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> </m:mrow> <m:mrow> <m:mi>x</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mo>−</m:mo> <m:mi>α</m:mi> <m:mrow> <m:mo stretchy="false">(</m:mo> <m:mrow> <m:mi>t</m:mi> </m:mrow> <m:mo stretchy="false">)</m:mo> </m:mrow> <m:mi>u</m:mi> <m:mo>,</m:mo> </m:mtd> </m:mtr> <m:mtr> <m:mtd columnalign="left"> <m:msub> <m:mrow> <m:mi>s</m:mi> </m:mrow> <m:mrow> <m:mi>t</m:mi> </m:mrow> </m:msub> <m:mo>=</m:mo> <m:mn>0</m:mn> <m:mo>.</m:mo> </m:mtd> </m:mtr> </m:mtable> </m:mrow> </m:mfenced> </m:math> $$\begin{cases}_{t}-{u}_{x}=0,\hfill \\ {u}_{t}+p{\left(v,s\right)}_{x}=-\alpha \left(t\right)u,\hfill \\ {s}_{t}=0.\hfill \end{cases}$$ Under some conditions on the initial data and the entropy function, we prove some existence theorems of global smooth solutions for the Cauchy problem. At present, there seems to be no research on this kind of problem.

Abstract This article investigates the existence and multiplicity of nontrivial solutions for Schrödinger-Bopp-Podolsky systems with critical nonlinearity in R 3 {{\mathbb{R}}}^{3} . Under appropriate assumptions about the potential and nonlinear terms, the existence and multiplicity of solutions are obtained by using the concentration-compactness principle and the symmetric mountain pass theorem. To some extent, we generalize the previous results.

Abstract This study investigates the conditions under which various curves maintain their characteristics when mapped between two regular surfaces in four-dimensional Euclidean space E 4 {{\mathbb{E}}}^{4} . We particularly focus on rectifying, osculating, and normal curves, examining their behavior under isometric, conformal, and homothetic conformal maps. By analyzing their behavior, this study reveals conditions under which these curves remain invariant when mapped between two surfaces. The findings contribute to a deeper understanding of geometric transformations in four-dimensional spaces.

Abstract We study the boundedness of commutators of the Hardy-Littlewood maximal function and the sharp maximal function on weighted Morrey spaces when the symbols of the commutators belong to weighted Lipschitz spaces (weighted Morrey-Campanato spaces). Some new characterizations for weighted Lipschitz functions are obtained in terms of the boundedness of the commutators.

Abstract This study investigates the renewal reward process under the assumption that the random variables describing the discrete interference of chance follow a generalized gamma distribution. A moment-based approximation method is employed to derive novel results for the renewal function, enabling an approximation of the ergodic distribution of the process. Furthermore, the limiting distribution of the ergodic distribution is also derived. The theoretical findings are illustrated through a specific example in which the demand random variable η 1 {\eta }_{1} is represented by a third-order Erlang distribution with parameter θ = 1 \theta =1 .

Abstract In this work, using the concepts of q q -calculus, we first define the q q -Jung-Kim-Srivastava and q q -Bernardi integral operators for multivalent functions. Then, we use these operators to establish the generalized integral operator ℬ q , p − m − λ f ( z ) {{\mathcal{ {\mathcal B} }}}_{q,p}^{-m-\lambda }f\left(z) for multivalent functions. By using the newly defined operator ℬ q , p − m − λ f ( z ) {{\mathcal{ {\mathcal B} }}}_{q,p}^{-m-\lambda }f\left(z) , we define a new class of multivalent analytic functions and diskuss various interesting features of the functions belonging to this class. Certain interesting examples of our developments are also considered. Furthermore, we explore the applications of the Carathéodory lemma in the form of specific results.

Abstract In this article, we provide a note on rigidity results for mean field-type equations with multiple exponential terms, including sinh-Gordon and Tzitzeica equations, on closed two-dimensional manifolds with positive Ricci curvature bounds. It is based on two nonlinear transformations and a geometric identity developed by Dolbeault et al.