Motivated by a recent work of Wang-Yang [19], we study the compactness of extremals $\left\{u_{\beta}\right\}$ for singular Hardy-Trudinger-Moser inequalities due to Hou [24]. In particular, by the method of blow-up analysis, we conclude that, up to a subsequence, $u_{\beta}$ converges to an extremal in some sense as β tends to zero.

This paper presents a method of lines solution based on the reproducing kernel Hilbert space method to the nonlinear one-dimensional Klein-Gordon equation that arises in many scientific fields areas. Our method uses discretization of the partial derivatives of the space variable to get a system of ODEs in the time variable and then solve the system of ODEs using reproducing kernel Hilbert space method. Consider two examples to validate the proposed method. Compare the results with the exact solution by calculating the error norms L₂ and L_∞ at various time levels. The results show that the presented scheme is a systematic, effective and powerful technique for the solution of Klein-Gordon equation.

In this paper, we obtain the blow-up result of solutions and some general decay rates for a quasilinear parabolic equation with viscoelastic terms

Due to the presence of the log source term, it is not possible to use the source term to dominate the term $A(t)\left|u_{t}\right|^{m-2} u_{t}$. To bypass this difficulty, we build up inverse Hölder-like inequality and then apply differential inequality argument to prove the solution blows up in finite time. In addition, we can also give a decay rate under a general assumption on the relaxation functions satisfying $g^{\prime} \leq-\xi(t) H(g(t))$, $H(t)=t^{v}$, $t \geq 0, v>1.$ This improves the existing results.

The purpose of this paper is to study scattering theory for the energy subcritical solutions to the non-radial defocusing inhomogeneous Hartree equation

Taking advantage of the decay factor in the nonlinearity instead of the embedding theorem, we establish the scattering criterion for the equation. Together with the Morawetz estimate, we obtain the scattering theory for the energy-subcritical case.

This paper considers the Cauchy problem of pseudo-parabolic equation with inhomogeneous terms $u_{t}=\Delta u+k \Delta u_{t}+w(x) u^{p}(x, t).$ In [1], Li et al. gave the critical Fujita exponent, second critical exponent and the life span for blow-up solutions under $w(x)=|x|^{\sigma}$ with σ>0. We further generalize the weight function $w(x) \sim|x|^{\sigma}$ for −2<σ<0, and discuss the global and non-global solutions to obtain the critical Fujita exponent.

We study the regularity and convergence of solutions for the n-dimensional (n=2,3) fourth-order vector-valued Helmholtz equations

for a given v in several Sobolev spaces, where β>0 and γ>0 are two given constants. By making use of the Fourier multiplier theorem, we establish the regularity and the $L^{p}-L^{q}$ estimates of solutions for Eq. (VFHE) under the condition $\mathbf{v} \in L^{p}\left(\mathbb{R}^{n}\right)$. We then derive the convergence that a solution u of Eq. (VFHE) approaches v weakly in $L^{p}\left(\mathbb{R}^{n}\right)$ and strongly in $L^{p}\left(\mathbb{R}^{n}\right)$ as the parameter pair (β,γ) approaches (0,0). In particular, as an application of the above results, for $(\mathbf{v}, \mathbf{u})$ solving the following viscous incompressible fluid equations

we gain the strong convergence in $L^{\infty}\left([T], L^{S}\left(\mathbb{R}^{n}\right)\right)$ from the Eqs. (VFHE)-(INS) to the Navier-Stokes equations as the parameter pair (β,γ) tending to (0,0), where $s=\frac{2 h}{h-2}$ with h>n.

This paper focuses on the statistical characteristics of the 3D MHD equations. We firstly establish an existence theorem of a Vishik-Fursikov measure of the 3D MHD equations by taking advantage of the Krein-Milman theorem along with some functional and measure theories. Then by applying the Topsoe lemma on the constructed trajectory space possessing some special topological properties, we show that the Vishik-Fursikov measure and the stationary Vishik-Fursikov statistical solution of the 3D MHD system are approximated by the counterparts of the 3D MHD-α system, respectively, as the parameter \alpha decreases to zero.