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 $\alpha =1$.

We consider the Cauchy problem for the nonlinear self-focusing Schrödinger equation in ℝ^N with an inhomogeneous smooth damping coefficient and we prove, for suitable initial data, and in the spirit of the seminal work of R. Glassey, a blowup result for the corresponding local solutions. We also give some lower bound estimates for the blowing-up solutions.

We consider the nonlinear Schrödinger equation with L²-critical exponent and an inhomogeneous damping term. By using the tools developed by Merle and Raphael, we prove the existence of blowup phenomena in the energy space H¹(ℝ).

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.

In this short paper, we focus on the blowup phenomenon of stochastic parabolic equations. We first discuss the probability of the event that the solutions keep positive. Then, the blowup phenomenon in the whole space is considered. The probability of the event that the solutions blow up in finite time is given. Lastly, we obtain the probability of the event that blowup time of stochastic parabolic equations larger than or less than the deterministic case.

We consider two new classes of examples of sup-norm blowup in finite time for strictly hyperbolic systems of conservation laws. The explosive growth in amplitude is caused either by a gradient catastrophe or by a singularity in the flux function. The examples show that solutions of uniformly strictly hyperbolic systems can remain as smooth as the initial data until the time of blowup. Consequently, blowup in amplitude is not necessarily strictly preceded by shock formation.

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.

In this paper, we study the blowup and non-global existence of smooth solutions to the one-dimensional Euler–Boltzmann equations of radiation hydrodynamics. First, we improve the blowup result in [P. Jiang and Y. G. Wang, Initial-boundary value problems and formation of singularities for one-dimensional non-relativistic radiation hydrodynamic equations, J. Hyperbolic Differential Equations 9 (2012) 711–738] on the half line $[0,+\infty )$ for large initial data by removing a restrict condition. Next, we obtain a new blowup result on the half line $[0,+\infty )$ by introducing a new momentum weight. Finally, we present two non-global existence results for the smooth solutions to the one-dimensional Euler–Boltzmann equations with vacuum on the interval $[0,1]$ by introducing some new average quantities.

In this paper, we study a chemotaxis system involving two interacting species as follows:

We prove the existence of solution to the problem whose behavior is concerned with reference to some systems through sup-sub-solution method.

This paper is concerned with a system of semilinear parabolic equations with two free boundaries describing the spreading fronts of the invasive species in a mutualistic ecological model. The local existence and uniqueness of a classical solution are obtained and the asymptotic behavior of the free boundary problem is studied. Our results indicate that two free boundaries tend monotonically to finite or infinite limits at the same time, and the free boundary problem admits a global slow solution with unbounded free boundaries if the intra-specific competitions are strong, while if the intra-specific competitions are weak, there exist the blowup solution and global fast solution.

In this paper, the long-time behavior and blowup of solutions for two generalized Ginzburg–Landau type equations with several nonlinear source terms are investigated. First, the conditions in which the solutions of the two equations are positive are analyzed by using Green’s function. According to the characteristics of nonlinear terms, four different functionals are constructed for solving the problems. Finally, we obtain the long-time behavior and finite-time blowup of solutions for the initial and boundary value problem by using these functionals.