In this paper, we consider a boundary control problem associated with a non-homogeneous pseudo-parabolic type equation in a bounded two-dimensional domain. In the part of the bound of the given region, the value of the solution is given, and it is required to find control to get the average value of the solution. The initial-boundary problem is solved by the Fourier method, and the control problem under consideration is analyzed with the Volterra integral equation of the second kind. The control function is found using the Laplace transform method and proved to be admissible.

This paper is concerned with a coupled system modeling the oscillations of suspension bridges, which consists of a beam equation and a viscoelastic string equation. We first transformed the original initial-boundary value problem into an equivalent one in the history space framework. Then we obtained the global well-posedness and regularity of mild solutions by using the semigroup theory. In addition, we employed the perturbed energy method to establish a stabilizability estimate. By verifying the gradient property and quasi-stability of the corresponding dynamical system, we derived the existence of a global attractor with finite fractal dimension.

In this paper, we considered quasilinear variational-hemivariational inequalities in the unbounded cylindrical domain $\mathbb{Q}=\mathbb{R}^{N} \times(0, \tau)$ of the form: Find u ∈ K ⊂ X with u(·, 0) = 0 satisfying $\left\langle u_{t}-\operatorname{div} A(x, t, \nabla u), v-u\right\rangle+\int_{\mathbb{Q}} a(x, t) j^{o}(x, t, u ; v-u) d x d t \geq 0, \forall v \in K,$ where K ⊂ X represents the bilateral constraints in $X=L^{p}\left(0, \tau ; D^{1, p}\left(\mathbb{R}^{N}\right)\right)$ with $D^{1, p}\left(\mathbb{R}^{N}\right)$ denoting the Beppo-Levi space (or homogeneous Sobolev space), and j^o(x,t,s; ϱ) denoting Clarke’s generalized directional derivative of the locally Lipschitz function s $\text { ↦ }$ j(x,t,s) at s in the direction ϱ. The main goal and the novelty of this paper was to prove existence results without assuming coercivity conditions on the time-dependent elliptic operators involved, and without supposing the existence of sub-supersolutions. Further difficulties arise in the treatment of the problem under consideration due to the lack of compact embedding of $\left.D^{1, p}\left(\mathbb{R}^{N}\right)\right)$ into Lebesgue spaces $L^{\sigma}\left(\mathbb{R}^{N}\right)$, and the fact that the constraint K has an empty interior, which prevents us from applying recent results on evolutionary variational inequalities. Instead our approach was based on an appropriately designed penalty technique and the use of weighted Lebesgue spaces as well as multi-valued pseudomontone operator theory.

We obtain a boundary pointwise gradient estimate on a parabolic half cube $Q_{2 R} \cap\left\{\left(x^{1}, x^{\prime}, t\right) \in\right.\left.\mathbb{R}^{n+1}: x^{1}>0\right\}$ for nonlinear parabolic equations with measurable nonlinearities, which are only assumed to be measurable in x¹-variable. The estimates are obtained in terms of Riesz potential of the right-hand side measure and the oscillation of the boundary data, where the boundary data is given on $Q_{2 R} \cap\left\{\left(x^{1}, x^{\prime}, t\right) \in \mathbb{R}^{n+1}: x^{1}=0\right\} .$

This paper studies a special 2D anisotropic incompressible Boussinesq equation in T² with T = [−1/2, 1/2] being a 1D periodic box. The system concerned here possesses vertical dissipation only in the vertical component of the velocity and vertical heat diffusion. When the buoyancy forcing is not present, the 2D Boussinesq equation is a 2D Navier-Stokes equation with vertical dissipation only in the vertical component. The stability and large-time behavior problem on the solutions to the 2D Navier-Stokes equation with only vertical or horizontal dissipation remains unknown. When coupled with the temperature, the global regularity to the system with vertical dissipation and vertical diffusion in R² has been solved by Cao and Wu (Arch. Ration. Mech. Anal., 208(2013), 985-1004). The stability with horizontal dissipation and horizontal diffusion in the periodic domain T × R has also been established by Dong, Wu, Xu, and Zhu (Calc. Var. Partial Differential Equations, 60(2021)) recently. Now whether the solution of the 2D system remains stable has yet to be solved when the velocity has vertical dissipation only in the u₂ equation. This paper aims to solve the problem and investigates the stability and large-time behavior of the solution to the special 2D Boussinesq equations on perturbations near the hydrostatic equilibrium. The basic idea here is to decompose the physical quantity f into its horizontal average, vertical average, and their corresponding oscillations. By establishing the strong Poincaré-type inequalities and several anisotropic inequalities related to the oscillations, we are able to obtain H²-stability of the solution under the assumptions that the initial data is sufficiently small and obeys some symmetries. Furthermore, the exponential decay rates for the oscillation parts in H¹ are also established.

By combining variational techniques with the saddle point theorem, we investigate the existence and nonexistence of periodic solutions to second-order partial difference equations involving p-Laplacians. Our obtained results generalize and complement some known ones. Finally, we display some examples and numerical simulations to show the validity of our main results.

In this paper, we consider the Cauchy problem of the three-dimensional isentropic compressible Navier-Stokes equations with degenerate viscosities. When the shear and bulk viscosity coefficients are both given as a constant multiple of the mass density’s power (ρ^δ with δ > 1), we show that the L^∞ norms of ∇u, ∇ρ^(γ−1)/2 and ∇ρ^(δ−1)/2 control the possible breakdown of classical solutions with far-field vacuum; this criterion is analogous to Serrin’s blowup criterion for the compressible Navier–Stokes equations.

We deal with the nonlinear Kirchhoff problem −(a + b ∫_R³ |∇u|²dx) Δu + V(x)u = f(u), x ∈ R³, (P) where a is a positive constant, b > 0 is a parameter, the potential function V is allowed to change its sign, and the nonlinearity f ∈ C(R, R) exhibits subcritical growth. Under some suitable conditions on V, we first prove that the problem has a positive ground state solution for all b > 0. Then, by using a more general global compactness lemma and a sign-changing Nehari manifold, combined with the method of constructing a sign-changing (PS)_c sequence, we show the existence of a least energy sign-changing solution for b > 0 that is sufficiently small. Moreover, the asymptotic behavior b ↘ 0 is established.

We provide a detailed analysis of the shock formation process for the non-isentropic 2d Euler equations in azimuthal symmetry. We prove that from an open set of smooth and generic initial data, solutions of the Euler equations form a first singularity or gradient blow-up. This first singularity is termed a Hölder C^1/3 pre-shock, and our analysis provides the first detailed description of this cusp solution. The novelty of this work relative to [1] is that we herein consider a much larger class of initial data, allow for a non-constant initial entropy, allow for a non-trivial sub-dominant Riemann variable, and introduce a host of new identities to avoid apparent derivative loss due to entropy gradients. The method of proof is also new and robust, exploring the transversality of the three different characteristic families to transform space derivatives into time derivatives. Our main result provides a fractional series expansion of the Euler solution about the pre-shock, whose coefficients are computed from the initial data.

This paper investigates the long-time behavior for a Navier–Stokes–Allen–Cahn system, a diffuse interface model for two-phase incompressible flows with unmatched densities, non-constant viscosities, and a singular Flory–Huggins potential. First, we establish the dissipativity of strong solutions via some a priori estimates. Then, we demonstrate the regular-continuity of the semigroup, which allows us to prove the existence of the global attractor in the strong solutions space.

This paper focuses on a class of fourth-order parabolic systems involving logarithmic and Rellich nonlinearities arising from modeling epitaxial thin film growth: $\left\{\begin{aligned} u_{t}+\Delta^{2} u & =|v|^{p}|u|^{p-2} u \ln |u v|-\mu \frac{u}{|x|^{4}}, \\ v_{t}+\Delta^{2} v & =|u|^{p}|v|^{p-2} v \ln |u v|-\gamma \frac{v}{|x|^{4}} . \end{aligned}\right.$ By using some new techniques to deal with the Rellich nonlinearities µ $\mu \frac{u}{|x|^{4}}$ and $\gamma \frac{v}{|x|^{4}}$, as well as the coupled logarithmic nonlinearities |v|^p|u|^p−2u ln |uv| and |u|^p|v|^p−2v ln |uv|, we prove the global existence and finite time blow-up of weak solutions. Furthermore, we not only obtain a new algebraic decay estimate and study the behavior of global weak solutions, but we also derive a new upper bound estimate for the blow-up time in case of the occurrence of blow-up.

In this paper, we established the boundedness of higher-order commutators I^m β, b generated by the fractional integral operator with BMO functions on grand weighted variable-exponent Herz- Morrey spaces M ˙K α, r),θ_λ,p(·)(ω). We also obtained the boundedness of the m−order multilinear fractional Hardy operator H_β,m and its adjoint operator H^∗ β, m on weighted variable-exponent Herz-Morrey spaces M ˙K α,λ q, p(·)(ω).