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In this paper, we consider the 1D Euler equation with time and space dependent damping term $-a(t,x)v$. It has long been known that when $a(t,x)$ is a positive constant or $0$, the solution exists globally in time or blows up in finite time, respectively. We prove that those results are invariant with respect to time and space dependent perturbations. We suppose that the coefficient $a$ satisfies the following condition $$ |a(t,x)- \mu_0| \leq a_1(t) + a_2 (x), $$ where $\mu_0 \geq 0$ and $a_1$ and $a_2$ are integrable functions with $t$ and $x$. Under this condition, we show the global existence and the blow-up with small initial data, when $\mu_0 >0$ and $\mu=0$ respectively., Comment: 18 pages. The global existence theorem is added. The statement and its proof of the blow-up theorem are simplified. The title is changed

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., Comment: 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

In this paper, we consider the 1D compressible Euler equation with the damping coefficient $\lambda/(1+t)^{\mu}$. Under the assumption that $0\leq \mu <1$ and $\lambda >0$ or $\mu=1$ and $\lambda > 2$, we prove that solutions exist globally in time, if initial data are small $C^1$ perturbation near constant states. In particular, we remove the conditions on the limit $\lim_{|x| \rightarrow \infty} (u (0,x), v (0,x))$, assumed in previous results., Comment: 9 pages. To appear in to appear in Advanced Studies in Pure Mathematics related to MSJ-SI 2018. arXiv admin note: text overlap with arXiv:1703.09821

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., Comment: 9 pages

The initial value problem for the two dimensional dissipative quasi-geostrophic equation derived from geophisical 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. In this paper, far field asymptotics of solutions are given in the critical and the supercritical cases.

In this paper, by using the theory of compensated compactness coupled with the kinetic formulation by Lions, Perthame, Souganidis and Tadmor \cite{LPT,LPS}, we prove the existence and nonexistence of global generalized (nonnegative) solutions of the nonlinearly degenerate wave equations $v_{tt} =c (|v|^{s-1} v)_{xx}$ with the nonnegative initial data $v_{0}(x)$ and $ s > 1$. This result is an extension of the results in the second author's paper \cite{Su}, where the existence and the nonexistence of the unique global classical solution were studied with a threshold on $\int_{-\infty}^{\infty} v_{1}(x) dx$ and the non-degeneracy condition $ v_{0}(x) \geq c_{0} > 0$ on the initial data., Comment: 15 pages

In this paper, we consider some blow-up problems for the 1D Euler equation 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 invariants. Furthermore, we give sharp lower and upper estimates of the lifespan of solutions, when initial data are small perturbations of constant states., Comment: 20 pages. The proof of Theorem 2.11(the global existence part) is modified

We consider the large time behavior of solutions to the following nonlinear wave equation: $\partial_{t}^2 u = c(u)^{2}\partial^2_x u + \lambda c(u)c'(u)(\partial_x u)^2$ with the parameter $\lambda \in [0,2]$. If $c(u(0,x))$ is bounded away from a positive constant, we can construct a local solution for smooth initial data. However, if $c(\cdot )$ has a zero point, then $c(u(t,x))$ can be going to zero in finite time. When $c(u(t,x))$ is going to 0, the equation degenerates. We give a sufficient condition that the equation with $0\leq \lambda < 2$ degenerates in finite time., Comment: 14 pages, 1 figure. Some typos are fixed. The proof in the case u_1 does not belong to L^1 added

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. When the exponent of the fractional Laplacian is large, large-time behavior of solutions is known. However, when the exponent is small, the perturbation methods used in the preceding works would not work. Large-time behavior of solutions to the drift-diffusion equation with small exponent is discussed. Particularly, the asymptotic expansion of solutions with high-order is derived.

We consider the global existence and blow up of solutions of the Cauchy problem of the quasilinear wave equation: $\partial_{t}^2 u = \partial_x(c(u)^2 \partial_x u)$, which has richly physical backgrounds. Under the assumption that $c(u(0,x))\geq \delta$ for some $\delta>0$, we give sufficient conditions for the existence of global smooth solutions and the occurrence of two types of blow-up respectively. One of the two types is that $L^{\infty}$-norm of $\partial_t u$ or $\partial_x u$ goes up to the infinity. The other type is that $c(u)$ vanishes, that is, the equation degenerates., Comment: 11 pages