$L^infty(Omega times (0,T))$ bound for heat solution
Let $Omega_T=(0,T) times Omega$, where $Omega$ a bounded smooth domain of $mathbb{R}^n$ and $T>0$. Let $ain L^infty(Omega)$ and consider the heat equation
$$u_t=Delta u + a(x)u, ;; (t,x)in Omega_T ,$$
$$u|_{partial Omega}=0,$$
$$u(0,cdot)=u_0.$$
Assume that the initial condition $u_0 in W^{2,infty}(Omega)$ (all derivative to order 2 are essentially bounded), can we prove that $uin L^infty(Omega times (0,T))$ and
$$|u_t|_{L^infty(Omega times (0,T))} leq C(|u_0|_{W^{2,infty}(Omega)}, |a|_infty).$$
If $a=0$ we can prove such result using heat kernel but for the equation with the potential I can't obtain the result.
If someone recommend a good reference it would be nice.
reference-request pde heat-equation semigroup-of-operators parabolic-pde
asked Dec 27 '18 at 18:01
S. Cho
464113
add a comment |
Let $Omega_T=(0,T) times Omega$, where $Omega$ a bounded smooth domain of $mathbb{R}^n$ and $T>0$. Let $ain L^infty(Omega)$ and consider the heat equation
$$u_t=Delta u + a(x)u, ;; (t,x)in Omega_T ,$$
$$u|_{partial Omega}=0,$$
$$u(0,cdot)=u_0.$$
Assume that the initial condition $u_0 in W^{2,infty}(Omega)$ (all derivative to order 2 are essentially bounded), can we prove that $uin L^infty(Omega times (0,T))$ and
$$|u_t|_{L^infty(Omega times (0,T))} leq C(|u_0|_{W^{2,infty}(Omega)}, |a|_infty).$$
If $a=0$ we can prove such result using heat kernel but for the equation with the potential I can't obtain the result.
If someone recommend a good reference it would be nice.
reference-request pde heat-equation semigroup-of-operators parabolic-pde
asked Dec 27 '18 at 18:01
S. Cho
464113
add a comment |
Let $Omega_T=(0,T) times Omega$, where $Omega$ a bounded smooth domain of $mathbb{R}^n$ and $T>0$. Let $ain L^infty(Omega)$ and consider the heat equation
$$u_t=Delta u + a(x)u, ;; (t,x)in Omega_T ,$$
$$u|_{partial Omega}=0,$$
$$u(0,cdot)=u_0.$$
Assume that the initial condition $u_0 in W^{2,infty}(Omega)$ (all derivative to order 2 are essentially bounded), can we prove that $uin L^infty(Omega times (0,T))$ and
$$|u_t|_{L^infty(Omega times (0,T))} leq C(|u_0|_{W^{2,infty}(Omega)}, |a|_infty).$$
If $a=0$ we can prove such result using heat kernel but for the equation with the potential I can't obtain the result.
If someone recommend a good reference it would be nice.
reference-request pde heat-equation semigroup-of-operators parabolic-pde
asked Dec 27 '18 at 18:01
S. Cho
464113
Let $Omega_T=(0,T) times Omega$, where $Omega$ a bounded smooth domain of $mathbb{R}^n$ and $T>0$. Let $ain L^infty(Omega)$ and consider the heat equation
$$u_t=Delta u + a(x)u, ;; (t,x)in Omega_T ,$$
$$u|_{partial Omega}=0,$$
$$u(0,cdot)=u_0.$$
Assume that the initial condition $u_0 in W^{2,infty}(Omega)$ (all derivative to order 2 are essentially bounded), can we prove that $uin L^infty(Omega times (0,T))$ and
$$|u_t|_{L^infty(Omega times (0,T))} leq C(|u_0|_{W^{2,infty}(Omega)}, |a|_infty).$$
If $a=0$ we can prove such result using heat kernel but for the equation with the potential I can't obtain the result.
If someone recommend a good reference it would be nice.
reference-request pde heat-equation semigroup-of-operators parabolic-pde
reference-request pde heat-equation semigroup-of-operators parabolic-pde
asked Dec 27 '18 at 18:01
S. Cho
464113
asked Dec 27 '18 at 18:01
S. Cho
464113
edited Dec 28 '18 at 23:33
asked Dec 27 '18 at 18:01
S. Cho
464113
asked Dec 27 '18 at 18:01
S. Cho
464113
asked Dec 27 '18 at 18:01
S. Cho
464113
464113
add a comment |
add a comment |
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