Narrow Results

We solve fundamental problems in Oka theory by establishing an implicit function theorem for sprays. As the first application of our implicit function theorem, we obtain an elementary proof of the fact that approximation yields interpolation. This proof and Lárusson’s elementary proof of the converse give an elementary proof of the equivalence between approximation and interpolation. The second application concerns the Oka property of a blowup. We prove that the blowup of an algebraically Oka manifold along a smooth algebraic center is Oka. In the appendix, equivariantly Oka manifolds are characterized by the equivariant version of Gromov’s condition ${\mathrm{\text{Ell}}}_1$, and the equivariant localization principle is also given.

This paper is concerned with the following radially symmetric Keller–Segel systems with nonlinear sensitivity $u_t=\mathrm{\Delta }u-\nabla \cdot (u{(1+u)}^{\alpha -1}\nabla v)$ and $0=\mathrm{\Delta }v-{⨍}_{\mathrm{\Omega }}u\mathrm{\text{d}}x+u$, posed on $\mathrm{\Omega }=\{x\in {ℝ}^n:|x|<R\}$$(n\ge 2)$ and subjected to homogeneous Neumann boundary conditions. It is well-known that $\frac{2}{n}$ is the critical exponent of the systems in the sense that all solutions exist globally if $\alpha <\frac{2}{n}$ and there exist finite-time blowup solutions if $\alpha >\frac{2}{n}$. Here we consider the supercritical case $\alpha \ge \frac{2}{n}$ and show a critical mass phenomenon. Precisely, we prove that there exists a critical mass $m_c:=m_c(n,R,\alpha )$ such that

• (1)for arbitrary nonincreasing nonnegative initial data $u_0(x)=u_0(|x|)$ with ${\int }_{\mathrm{\Omega }}{u}_0>m_c$ and $u_0\not\equiv {⨍}_{\mathrm{\Omega }}{u}_0$, the corresponding solution blows up in finite time if $\alpha >\frac{2}{n}$, and if $\alpha =\frac{2}{n}$ we can only prove that the solution blows up in finite time or infinite time;
• (2)for some nonincreasing nonnegative initial data with ${\int }_{\mathrm{\Omega }}{u}_0<m_c$, the corresponding solutions are globally bounded.

Our results extend that of Winkler’s paper [M. Winkler, How unstable is spatial homogeneity in Keller–Segel systems? A new critical mass phenomenon in two- and higher-dimensional parabolic–elliptic cases, Math. Ann. 373 (2019) 1237–1282], where he proved similar results for the system with $\alpha =1$.

This paper is concerned with complex-valued solutions of the Korteweg–de Vries equation. Interest will be focused upon the initial-value problem with initial data that is periodic in space. Derived here are results of local and global well-posedness, singularity formation in finite time and, perhaps surprisingly, results of non-existence. The overall picture is notably different from the situation that obtains for real-valued solutions.

We are concerned with the global existence and blowup of the classical solutions to the Cauchy problem of one-dimensional isentropic relativistic Euler equations (Chaplygin gas, pressureless perfect fluid and stiff matter) with linearly degenerate characteristics. We at first derive the exact representation formula for all the fluids by the property of linearly degenerate. Then for the Chaplygin gas and the pressureless perfect fluid, we give a classification of the initial data that leads to the global existence and the blowup of the classical solution, respectively. We construct, especially, a class of initial data that contributes to the formation of “cusp-type” singularity and study the evolution of the solution after blowup by introducing a weak solution called delta shock wave. At last, for the stiff matter, we show that this system is indeed a linear system and prove the global existence of the classical solution to this fluid.