Motivated by the mean field equations with probability measure derived by Sawada-Suzuki and by Neri in the context of the statistical mechanics description of two-dimensional turbulence, we study the semilinear elliptic equation with probability measure: \begin{equation*} -\Delta v=\lambda\int_IV(\alpha,x,v)e^{\alpha v}\,{\mathcal P(d\alpha)} -\frac{\lambda}{|\Omega|}\iint_{I\times\Omega}V(\alpha,x,v)e^{\alpha v}\,{\mathcal P(d\alpha)} dx, \end{equation*} defined on a compact Riemannian surface. This equation includes the above mentioned equations of physical interest as special cases. For such an equation we study the blow-up properties of solution sequences. The optimal Trudinger-Moser inequality is also considered.

We study the degenerate parabolic equation with critical exponent derived from the kinetic theory using R\'enyi-Tsallis entropy, and show an $\varepsilon$-regularity theorem concerning the local uniform bound of the solution.

We study an interacting system of chemotactic species in two space dimensions. First, we show that there is a parameter region which ensures simultaneous blowup also for non-radially symmetric solutions. If the existence time of the solution is finite, there is a formation of collapse (possibly degenerate) for each component, total mass quantization, and formation of subcollapses. For radially symmetric solutions we can rigorously prove that the collapse concentrates mass on one component if the total masses of the other components are relatively small. Several related results are also shown.

In this paper we study a strain-gradient type thermoviscoelastic system. We focus on the stationary states and their dynamical stability. The adiabatic stationary state is formulated as a nonlinear eigenvalue problem with non-local terms associated with the total energy conservation. One of the purposes of this paper is to extend the results obtained in Suzuki-Tasaki [34]. We reveal a unified structure, called semi-dualities, of the thermoviscoelastic system of viscosity-capillarity type with temperature-dependent viscous and elastic moduli. We describe a physical background and outline the thermodynamic derivation of the system. Based on the semi-dual structure we construct a series of general results concerning the stationary states and their stability. The application of these results together with the bifurcation theory allows us to analyze the total set of the stationary solutions in more detail.

The aim of this paper is to investigate the asymptotic behavior of solutions to the limit unstable Cahn-Hilliard equation with inertial term $$ \partial_t^2 u + \partial_t u + \Delta ( \Delta u + | u |^{p-1} u)=0, \quad {\rm in } \quad (0,\infty) \times {\mathbb{R}}^n. $$ We shall prove the unique global existence of a small solution for the equation in the super critical case $p > 1 + \frac{2}{n}$ in $n=1,2,3$, and also give an asymptotic profile of the solution. By considering the higher-order expansion of the solution, we obtain more precise information about optimal decay of the solution under the more restricted condition $p > 1+ \frac{4}{n}$, and observe the contribution of the nonlinear term to the solution.

In this paper we prove the blow-up theorem in the critical case for weakly coupled systems of semilinear wave equations in high dimensions. The upper bound of the lifespan of the solution is precisely clarified.

This paper is concerned with a focusing nonlinear Schrödinger equation whose nonlinearity consists of the energy-critical local interaction term with a perturbation of the $L^2$-super and energy-subcritical term. We prove the existence of a ground state (= a standing-wave solution of minimal action) when the space dimension is four or higher and prove the nonexistence of any ground state when the space dimension is three and the perturbation is small. Once we have a ground state, a so-called potential-well scenario works well, so that we can give a sufficient condition for the nonexistence of global-in-time solutions.

We deduce Liouville type theorems for the Navier-Stokes and the Euler equations on ${\mathbb R}^N$, $N\geq 2$. Specifically, we prove that if a weak solution $(v,p)$ satisfies $|v|^2 +|p| \in L^1 (0,T; L^1({\mathbb R}^N, w_1(x)dx))$ and $\int_{{\mathbb R}^N} p(x,t)w_2 (x)dx \geq0$ for some weight functions $w_1(x)$ and $w_2 (x)$, then the solution is trivial, namely $v=0$ almost everywhere on ${\mathbb R}^N \times (0, T)$. Similar results hold for the MHD equations on ${\mathbb R}^N$, $N\geq3$.