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We study the Cauchy problem for the quasilinear wave equation $ \partial^2 _t u = u^{2a} \partial^2_x u + F(u) u_x $ with $a \geq 0$ and show a result for the local in time existence under new conditions. In the previous results, it is assumed that $u(0,x) \geq c_0>0$ for some constant $c_0$ to prove the existence and the uniqueness. This assumption ensures that the equation does not degenerate. In this paper, we allow the equation to degenerate at spacial infinity. Namely we consider the local well-posedness under the assumption that $u(0,x)>0$ and $u(0,x) \rightarrow 0$ as $|x| \rightarrow \infty$. Furthermore, to prove the local well-posedness, we find that the so-called Levi condition appears. Our proof is based on the method of characteristic and the contraction mapping principle via weighted $L^\infty$ estimates., 20 pages

We consider the blow-up of solutions to the following parameterized nonlinear wave equation: $ u_{tt} = c(u)^{2} u_{xx} + \lambda c(u)c'(u)( u_x)^2$ with the real parameter $\lambda$. In previous works, it was reported that there exist finite time blow-up solutions with $\lambda =1$ and $2$. However, the construction of a blow-up solution depends on the symmetric structure of the equation (e.g., the energy conservation law). In the present paper, we extend the blow-up result with $\lambda =1$ to the case with $\lambda \in (0,1]$ by using a new $L^{2/\lambda}$ estimate. Moreover, some properties for the blow-up solution including the H\"older continuity are also discussed., Comment: accepted by Indiana University Mathematics Journal in 2021

We study the drift-diffusion equation with fractional dissipation $$(-\varDelta )^{\theta /2}$$ arising from a model of semiconductors. First, we prove the existence and uniqueness of the small solution to the corresponding stationary problem in the whole space. Moreover, it is proved that the unique solution of non-stationary problem exists globally in time and decays exponentially, if initial data are suitably close to the stationary solution and the stationary solution is sufficiently small.

In this paper, we consider some blow-up problems for the 1D Euler equations with time and space dependent damping. We investigate sufficient conditions on initial data and the rate of spatial or time-like decay of the coefficient of damping for the occurrence of the finite time blow-up. In particular, our sufficient conditions ensure that the derivative blow-up occurs in finite time with the solution itself and the pressure bounded. Our method is based on simple estimates with Riemann invariant. Furthermore, we give sharp lower and upper estimates of the lifespan of solutions, when initial data are small perturbations of constant states.

The initial value problem for the two dimensional dissipative quasi-geostrophic equation of the critical and the supercritical cases is considered. Anomalous diffusion on this equation provides slow decay of solutions as the spatial parameter tends to infinity. In this paper, uniform estimates for far field asymptotics of solutions are given., 9 pages

The initial-value problem for the drift–diffusion equation arising from the model of semiconductor device simulations is studied. The dissipation on this equation is given by the fractional Laplacian ( − Δ ) θ / 2 . Large-time behavior of solutions to the drift–diffusion equation with 0 θ ≤ 1 is discussed. When θ > 1 , large-time behavior of solutions is known. However, when 0 θ ≤ 1 , the perturbation methods used in the preceding works would not work. In this paper, the asymptotic expansion of solutions with high-order is derived.

The initial value problem for the two-dimensional dissipative quasi-geostrophic equation derived from geophysical fluid dynamics is studied. The dissipation of this equation is given by the fractional Laplacian. It is known that the half Laplacian is a critical dissipation for the quasi-geostrophic equation. The global existence of solutions upon the suitable condition is also well known, and that solutions of a fractional dissipative equation decay with a polynomial order as the spatial variable tends to infinity. In this paper, far field asymptotics of solutions to the quasi-geostrophic equation are given in the critical and the supercritical cases. Those estimates are derived from the energy methods for the difference between the solution and its asymptotic profile.

In this paper, the initial value problem for the drift-diffusion equation which stands for a model of a semiconductor device is studied. When the dissipative effect on the drift-diffusion equation is given by the half Laplacian, the dissipation balances to the extra force term. This case is called critical. The goal of this paper is to derive decay and asymptotic expansion of the solution to the drift-diffusion equation as time variable tends to infinity.

We study local and global existence and uniqueness of solutions to the drift-diffusion equation with fractional dissipation ( − Δ ) θ / 2 . In the preceding works for some associated equations, the cases θ = 1 and θ 1 are known as critical and supercritical respectively. In the critical and supercritical cases, we may not apply the L p -theory for semilinear equations of parabolic type used in the subcritical case 1 θ ≤ 2 . We discuss local existence with large data and global existence with small data in the Besov space B p , q n p − θ ( R n ) , which corresponds to the scaling invariant space of the equation. Furthermore we show that solutions can blow up in finite time if initial data is not small.

We study the local-in-time existence and uniqueness of the Cauchy problem for the nonlinear wave equation [Formula: see text], which is called the second sound equation. Assuming that u(0, x) = φ ≥ A > 0, φ ∈ C1, and ∂xφ ∈ Hs, we establish the uniqueness of solutions without restriction on their amplitude.