Browsing by Author "Nakao, Mitsuhiro"

Now showing 1 - 4 of 4

Access status: Open Access ,
Existence and decay of finite energy solutions for semilinear dissipative wave equations in time-dependent domains
(Wydawnictwa AGH, 2020) Nakao, Mitsuhiro
We consider the initial-boundary value problem for semilinear dissipative wave equations in noncylindrical domain $\bigcup_{0\leq t \lt\infty} \Omega(t)\times\{t\} \subset \mathbb{R}^N\times \mathbb{R}$. We are interested in finite energy solution. We derive an exponential decay of the energy in the case $\Omega(t)$ is bounded in $\mathbb{R}^N$ and the estimate $\int\limits_0^{\infty} E(t)dt \leq C(E(0),\|u(0)\|)\lt \infty$ in the case $\Omega(t)$ is unbounded. Existence and uniqueness of finite energy solution are also proved.
Access status: Open Access ,
Existence and smoothing effects of the initial-boundary value problem for ∂u/∂t−Δσ(u)=0 in time-dependent domains
(Wydawnictwa AGH, 2023) Nakao, Mitsuhiro
We show the existence, smoothing effects and decay properties of solutions to the initial-boundary value problem for a generalized porous medium type parabolic equations of the form $u_t-\Delta \sigma(u) =0 \quad \text{in } Q(0, T)$ with the initial and boundary conditions $u(0)=u_0 \quad \text{and} \quad u(t)|_{\partial \Omega(t)}=0,$ where $\Omega(t)$ is a bounded domain in $R^N$ for each $t \geq 0$ and $Q(0,T)=\bigcup_{0 \lt t \lt T} \Omega(t) \times \{t\}, \quad T>0.$ Our class of $\sigma(u)$ includes $\sigma(u)=|u|^m u$, $\sigma(u)=u \log (1+ |u|^m)$, $0 \leq m \leq 2$, and $\sigma(u)=|u|^{m}u/\sqrt{1+|u|^2}$, $1 \leq m \leq 2$, etc. We derive precise estimates for $\|u(t)\|_{\Omega(t),\infty}$ and $\|\nabla\sigma(u(t))\|^2_{\Omega(t),2}$, $t \gt 0$, depending on $\|u_0\|_{\Omega(0),r}$ and the movement of $\partial\Omega(t)$.
Access status: Open Access ,
Global solutions to the initial-boundary value problem for the quasilinear viscoelastic equation with a derivative nonlinearity
(2014) Nakao, Mitsuhiro
We prove the existence and uniqueness of a global decaying solution to the initial boundary value problem for the quasilinear wave equation with Kelvin-Voigt dissipation and a derivative nonlinearity. To derive the required estimates of the solutions we employ a ‘loan’ method and use a difference inequality on the energy.
Access status: Open Access ,
Remarks on global solutions to the initial-boundary value problem for quasilinear degenerate parabolic equations with a nonlinear source term
(Wydawnictwa AGH, 2019) Nakao, Mitsuhiro
We give an existence theorem of global solution to the initial-boundary value problem for $u_{t}-\operatorname{div}\{\sigma(|\nabla u|^2)\nabla u\}=f(u)$ under some smallness conditions on the initial data, where $\sigma (v^2)$ is a positive function of $v^{2}\neq 0$ admitting the degeneracy property $\sigma(0)=0$. We are interested in the case where $\sigma(v^{2})$ has no exponent $m \geq 0$ such that $\sigma(v^2) \geq k_0|v|^m , k_0 \gt 0$. A typical example is $\sigma(v^2)=\operatorname{log}(1+v^2)$. $f(u)$ is a function like $f=|u|^{\alpha} u$. A decay estimate for $\|\nabla u(t)\|_{\infty}$ is also given.