We show how a single Keller-Segel model with rotational flux terms can be used to answer the questions of global existence and blow-up of solutions for several others Keller-Segel-type models arising in mathematical biology and physics. In the case of aggregation of two species on a two-dimensional domain, we identify a threshold curve in the plane of masses that allows us to determine whether the solution of the system blows up or is global in time. As a byproduct, our research provides a novel blow up result for a mathematical model recently introduced in the literature describing the dynamics of two types of Brownian vortices with different signs.

This paper investigates the initial boundary value problem (IBVP) associated with the fifth-order Korteweg-de Vries (KdV) equation on a finite interval: \begin{equation} \begin{cases} {\partial _t} u+{\partial _x} u+\beta {\partial _x} ^3u +{\partial _x} ^5u= G(u), \ \ 0 < x < L , \\ u(x,0)=\phi(x), \end{cases}\tag{0.1} \end{equation} subject to non-homogeneous boundary conditions: \begin{equation} B_j u = h_j(t), \quad j = 1,2,3,4,5, \quad t > 0, \tag{0.2} \end{equation} where $ G(u)= c_0 u{\partial _x} u + c_1{\partial _x} u{\partial _x}^2 u + c_2 {\partial _x} (u{\partial _x}^2 u) + c_3 {\partial _x} (u^3), $ and $$ B_j u = \sum_{k=0}^{4} (a_{jk} \partial_{x}^{k}u(0,t) + b_{jk} \partial_{x}^{k}u(L,t)), \ j=1, \cdots, 5, $$ and $a_{jk}, b_{jk}$ are real constants for $k=0,1,2,3,4$ and $j=1,2,3,4,5$. Under certain general assumptions on these coefficients, the well-posedness of the IBVP $(0.1)-(0.2)$ is established, showing local analytical well-posedness in the space $H^{s}(0,L)$ for $s \geq 1$ (and for some specified cases, $s \geq 0$), with initial data $\phi \in H^{s}(0,L)$ and boundary values $h_j$ belonging to appropriate spaces with optimal regularity. In contrast, for the pure initial value problem (IVP) of the fifth-order KdV equation posed on $\mathbb R$, it is known to be continuous ($C^0$) well-posed in the space $H^s (\mathbb{R})$ for $s \geq 2$, but it fails to be analytically well-posed in $H^s (\mathbb{R})$ for any $s \in \mathbb{R}$. The remarkable strong Kato smoothing and double sharp Kato smoothing properties, attributed to Kenig, Ponce, and Vega [42, 45], of the pure initial value problem (IVP) of the linear inhomogeneous fifth order KdV equation posed on the whole line $\mathbb{R}$, $$ {\partial _t} v +\beta {\partial _x}^3 v + {\partial _x} ^5 v = g (x,t), \quad v(x,0) =0, \ x, \ t \, \in \mathbb{R}, $$ have played a pivotal role in establishing the well-posedness of the IBVP $(0.1)-(0.2)$.

Our purpose is to consider the life span of solutions to the nonlinear Neumann boundary value problem for one dimensional nonlinear Schrödinger equation on a half-line \begin{equation*} \begin{cases} i\partial _{t}u+\tfrac{1}{2} \partial _{x}^{2}u=0, & t > 0, x\in \mathbb{R}_{+}, \\ u(0,x)=u_{0}(x), & x \in \mathbb{R}_{+}, \\ -\partial _{x}u( t,0) = g( t,0) , & t > 0, \end{cases} \end{equation*} where $g( t,0) =e^{-\frac{\pi }{4}i} | u ( t,0 ) | ^{q}.$ We prove that for any $q > 2$, and $r > 2( q-1) $, there exists an initial function $u_{0}\in \dot{H}_{x}^{- ( \frac{2}{r}-\frac{1}{2} ) }$ such that the maximal existence time is finite.

We consider 3D incompressible non-Newtonian fluid, subject to a dynamic boundary condition. Using an iteration scheme in Nikolski-Bochner spaces, we obtain additional fractional time regularity of arbitrary weak solution, provided the power-law exponent is above the critical value $r = 11/5.$ This implies uniqueness of solutions. We also show existence of the global attractor and even a finite-dimensional exponential attractor.

This article deals with the existence of solutions for a class of modified Schrödinger-Poisson system with $p$-Laplacian and critical nonlinearity in $\mathbb{R}^{3}$. First, by using variable transformations, the quasilinear equation is reduced to a semilinear equation, whose corresponding functional is well defined in the classical Sobolev space and satisfies the geometric conditions of mountain pass theorem. Next, under suitable assumptions on potential functions and nonlinear terms, we can prove the compactness condition by using the concentration-compactness principle which is completely different from previous methods of proving compactness conditions. Finally, some existence results of solutions for this problem are obtained with the help of the variational method. In a sense, we generalize and fill in some of the previous results [17, 24, 32, 33].