论文标题
各向异性抛物线方程的存在和全球二阶规则性具有可变增长
Existence and global second-order regularity for anisotropic parabolic equations with variable growth
论文作者
论文摘要
我们考虑各向异性抛物线方程\ [u_t- \ sum_ {i = 1}^nd_ {x_i} \ left（| d_ {x_i} u |^i |^i |^i | | $ω\ times（0，t）$，其中$ω\ subset \ mathbb {r}^n $，$ n \ geq 2 $是平行教。非线性$ p_i $的指数具有Lipschitz连续功能。显示的是，如果$ p_i（x，t）> \ frac {2n} {n+2} $，\ [μ= \ sup_ {q_t} \ dfrac {\ max_i p_i p_i（x，x，x，t）} | d_ {x_i} u_0 |^{\ max \ {p_i（\ cdot，0），2 \}}} \ in l^1（ω），\ quad f \ in l^2（0，t; w^{1,2}} _0（ω），\ w^{1,2} _0（ω），\] c（[0，t]; l^2（ω））$带有$ | d_ {x_i} u |^{p_i} \ in L^{\ infty}（0，t; t; l^1（ω）$，$ u_t \ in L^2（q_t）$。此外，\ [| d_ {x_i} u |^{p_i+r} \在l^1（q_t）\ quad \ text {带有$ r = r = r（μ，n）> 0 $}，\ qquad | quad | d_i} w^{1,2}（q_t）。 \]如果$ q_t $的横向边界上的$ p_i = 2 $，则声明对于平滑域$ω$仍然是正确的。
We consider the homogeneous Dirichlet problem for the anisotropic parabolic equation \[ u_t-\sum_{i=1}^ND_{x_i}\left(|D_{x_i}u|^{p_i(x,t)-2}D_{x_i}u\right)=f(x,t) \] in the cylinder $Ω\times (0,T)$, where $Ω\subset \mathbb{R}^N$, $N\geq 2$, is a parallelepiped. The exponents of nonlinearity $p_i$ are given Lipschitz-continuous functions. It is shown that if $p_i(x,t)>\frac{2N}{N+2}$, \[ μ=\sup_{Q_T}\dfrac{\max_i p_i(x,t)}{\min_i p_i(x,t)}<1+\dfrac{1}{N}, \quad |D_{x_i}u_0|^{\max\{p_i(\cdot,0),2\}}\in L^1(Ω),\quad f\in L^2(0,T;W^{1,2}_0(Ω)), \] then the problem has a unique solution $u\in C([0,T];L^2(Ω))$ with $|D_{x_i} u|^{p_i}\in L^{\infty}(0,T;L^1(Ω))$, $u_t\in L^2(Q_T)$. Moreover, \[ |D_{x_i}u|^{p_i+r}\in L^1(Q_T)\quad \text{with some $r=r(μ,N)>0$},\qquad |D_{x_i}u|^{\frac{p_i-2}{2}}D_{x_i}u\in W^{1,2}(Q_T). \] The assertions remain true for a smooth domain $Ω$ if $p_i=2$ on the lateral boundary of $Q_T$.
