We deal with (KS)$_m$ below for the super-critical cases of $q \ge m+\frac{2}{N}$ with $N \ge 2, \ m \ge 1, \ q \ge 2$. Based on an $\varepsilon$-regularity theorem in [20], we prove that the set $S_u$ of blow-up points of the weak solution $u$ consists of finitely many points if $u^{\frac{N(q-m)}{2}} \in C_w([0,T]; L^1(\mathbb R^N))$. Moreover, we show that $u^{\frac{N(q-m)}{2}}$ forms a delta-function singularity at the blow-up time. Simultaneously, we give a sufficient condition on $u$ such that $u^{\frac{N(q-m)}{2}} \in C_w([0,T]; L^1(\mathbb R^N))$. Our condition exhibits a scaling invariant class associated with (KS)$_m$.

We consider the effects on a blow-up phenomena of the Keller-Segel system (KS) in terms of the mass and second moment of initial data in connection with three coefficients $\gamma, \alpha, \chi$. In particular, for $\gamma=0,$ our criterion on blow-up of solutions coincides with the quantity of the scaling invariant class associated with the Keller-Segel system. We also show that the size of the $L^{\frac{N}{2}}$-norm plays an important role in construction of the time global and blow-up solutions of (KS). Furthermore, we give essential examples of small-$L^1$ initial data which yield blow-up solutions. Consequently, we give the answer to the conjecture by Childress-Percus [2] for $N \ge 3$; i.e., that even though the $L^1$-norm of the initial data is small, the blow-up solutions of (KS) exist in the case of $N \ge 3$. This implies that the smallness of the $L^1$-norm of the initial data does not give us any criterion on the existence of global solutions except when $N=2$.

We consider the Cauchy problem of the semilinear damped wave system: \begin{equation} \notag \begin{cases} \partial_{t}^2 u_{j} - \Delta u_{j} + \partial_{t} u_{j} = F_{j}(u), & t > 0, \quad x\in \mathbb R^{n},\\ u_{j}(0,x)=a_{j}(x),\quad \partial_{t} u_{j}(0,x) = b_{j}(x), & x\in \mathbb R^{n}, \end{cases} \end{equation} where $m \ge 2$ and $j = 1$, $\cdots$, $m$. We show the existence of a global-in-time solution for a small initial data under a sharp condition on the nonlinear exponents, which is a natural extension of the results for the single nonlinear damped wave equations ([28], [30]). The proof is based on $L^{p}$-$L^{q}$ type estimates of the fundamental solutions of the linear damped wave equations ([9]) and systematic choice of the function scale to adjust the nonlinear growth order.

We prove that every eigenvalue of a Robin problem with boundary parameter $\alpha$ on a sufficiently smooth domain behaves asymptotically like $-\alpha^2$ as $\alpha \to \infty$. This generalizes an existing result for the first eigenvalue.

We consider the Cauchy problem for the following reaction-diffusion system: $$ \begin{cases} \displaystyle \frac{\partial u_i}{\partial t} = \Delta u_i +g_i(x,t) \prod_{j=1}^m {u_j}^{p_{ij}}, & x \in {\bf R}^n, \ t > 0, \ i=1,2,\cdots,m, \\ u_i(x,0)=f_i(x) \geq 0, \ \not\equiv 0, & x \in {\bf R}^n, \ i=1,2,\cdots,m, \end{cases} $$ where $n \geq 3$, $m \geq 2$, $p_{ij} \geq 0$ $( 1 \leq i, j \leq m ),$ $ \prod_{j=1}^m {u_j}^{p_{ij}} = {u_1}^{p_{i1}} {u_2}^{p_{i2}} \cdots $ ${u_m}^{p_{im}} ,$ $ (i=1,2,\cdots,m) $ and $f_i(x)$ ($i=1,2,\cdots,m$) is a non-negative, bounded and continuous function in ${\bf R}^n$. In this paper, we show the existence of non-negative and global solutions $u_i(x,t)$ ($i=1,2,\cdots,m$) for the above Cauchy problem when $g_i(x,t)$ ($i=1,2,\cdots,m$) and $p_{ij} \geq 0$ ($1 \leq i, j \leq m$) satisfy some conditions.

An elliptic boundary problem with a solution-dependent measure is studied. Applications to elliptic transmission problems with a free interface and to a generalized elliptic obstacle problem with a nonzero Neumann term on the free boundary are given.

We consider the system $$ \begin{cases} -\varepsilon^{2} \Delta u +W(x)u=Q_{u}(u,v)+ \frac{1}{2^*}K_u(u,v)~\text{in } \mathbb{R}^N, \\ -\varepsilon^{2} \Delta v +V(x)v=Q_{v}(u,v)+ \frac{1}{2^*}K_v(u,v)~\text{in } \mathbb{R}^N, \\ u,v \in H^{1}(\mathbb{R}^N),u(x),v(x)>0~~\text{for each } x \in \mathbb{R}^N, \end{cases} $$ where $2^*=2N/(N-2)$, $N \geq 3$, $\varepsilon>0$ is a parameter, $W$ and $V$ are positive potentials, and $Q$ and $K$ are homogeneous function with $K$ having critical growth. We relate the number of solutions to the topology of the set where $W$ and $V$ attain their minimum values. In the proof, we apply Ljusternik-Schnirelmann theory.

We consider the initial-boundary-value problem for a nonlinear pseudodifferential equation on a segment. We are interested in the case of a nonanalytic symbol of the pseudodifferential operator $K(p)=\sqrt{\left\vert p\right\vert }.$ We study traditionally important problems of the theory of nonlinear partial differential equations, such as global-in-time existence of solutions to the initial-boundary-value problem and the asymptotic behavior of solutions for large time.

We demonstrate the large deviation property for the mild solutions of stochastic evolution equations with monotone nonlinearity and multiplicative noise. This is achieved using the recently developed weak convergence method, in studying the large deviation principle. An Itô-type inequality is a main tool in the proofs. We also give two examples to illustrate the applications of the theorems.

The aim of this work is to investigate the asymptotic behavior of solutions near hyperbolic equilibria for nonlinear partial neutral functional differential equations. We suppose that the linear part $A$ satisfies the Hille-Yosida condition on a Banach space and is not necessarily densely defined; the delayed part is assumed to be Lipschitz. We show the existence of stable and unstable manifolds near hyperbolic equilibria when the neutral operator is stable and the semigroup generated by the part of $A$ in $\overline{D(A)}$ is compact. Local stable and unstable manifolds are also obtained when the undelayed part is a C$^{1}$ function in a neighborhood of the equilibria.

Here the authors want to point out that the term $I_2,$ of Proposition 2 of the original paper ([2]), has to be estimated in a different way. Moreover, now the proof of Theorem 4 of the original paper holds with the new assumption (2) (involving $K_w.3)'$ mentioned below), instead of (6.2) of [2] (involving $K_w.3)$ of the original paper), while, since Theorem 3 (convergence theorem) can be proved with both assumptions $K_w.3)$ and $K_w.3)'$, we prefer here to use directly $K_w.3)'$, in analogy with condition (2). Let us notice that it is easy to see that the two conditions $K_w.3)$ and $K_w.3)'$ cannot be compared. Here we want also to point out that in the convergence theorem of [3] as well as in Lemma 2 of [3], a similar problem occurs and it is solved in the same way proving that $V_{\varphi}[\lambda (H_w \circ f-f)]{\mbox{$\rightarrow$}} 0$, as $w{\mbox{$\rightarrow$}} +\infty$ for sufficiently small $\lambda>0,$ using assumption $K_w.3)$ (see Remark below).