Parabolic equations with \(p,q\)-growth
Given \(\Omega\subset\mathbb R^N\) a bounded open set with boundary \(\partial\Omega\) and \(T>0\), the authors consider the parabolic problem NEWLINE\[NEWLINE \partial_t u-\mathrm{div}a(x,t,\nabla u)=0\text{ in }\Omega_T=\Omega\times(0,T). \tag{1}NEWLINE\]NEWLINE The vector field \(a:\Omega_T\times\mathbb R^N\to \mathbb R^N\) is assumed to be differentiable with respect to the spacial variable \(x\) and with respect to the gradient variable, and to satisfy the following non-standard growth and ellipticity conditions: NEWLINENEWLINENEWLINE\[NEWLINE\begin{cases} | a(x,t,\xi )| +(\mu^2+|\xi|^2)^{\frac {1}{2}}|\partial_\xi a(x,t,\xi)|\leq L(\mu^2+|\xi|^2)^{\frac{q-1}{2}},\\ \langle \partial_\xi a(x,t,\xi)\tilde\xi,\tilde\xi\rangle \geq \nu(\mu^2+| \xi|^2)^{\frac{p-2}{2}}|\tilde\xi|^2,\\ |\partial_xa_j(x,t,\xi)|\leq L(\mu^2+|\xi|^2)^{\frac{p+q-2}{4}},\\ |\partial_{\xi_l}a_j(x,t,\xi)-\partial_{\xi_j}a_l(x,t,\xi)|\leq L(\mu^2+|\xi|^2)^{\frac{p+q-4}{4}}, \end{cases}\tag{2}NEWLINE\]NEWLINENEWLINENEWLINEfor any \((x,t)\in\Omega_T\), \(\xi,\tilde\xi\in\mathbb R^N\), \(j,l=1,\ldots ,N\), where \(2\leq p<q\) and \(\nu, L>0\) and \(\mu\in [0,1]\).NEWLINENEWLINEA weak solution to (1) is defined to be a function \(u\in L_{\mathrm{loc}}^q((0,T); W_{\mathrm{loc}}^{1,q}(\Omega))\) such that for every \(\varphi\in C_0^\infty(\Omega_T)\), NEWLINE\[NEWLINE \int_{\Omega_T}(u\cdot\varphi_t-\langle a(x,t,\nabla u),\nabla\varphi\rangle)dz=0.\tag{3}NEWLINE\]NEWLINE Under the assumption (2) and for \(2\leq p\leq q<p+\frac{4}{N}\), they show that any weak solution \(u\) of Problem (1) satisfies the following: \(\nabla u\in L_{\mathrm{loc}}^\infty(\Omega_T,\mathbb R^N)\) and for any parabolic cylinder \(Q_\rho (z_0)\Subset \Omega_T\) and \(s\in (0,1)\) there holds NEWLINENEWLINENEWLINE\[NEWLINE\begin{aligned} \sup_{Q_{s\rho}(z_0)}|\nabla u|\leq C\left[\frac{1}{[(1-s)\rho]^{\hat N+2}}\int_{Q_\rho(z_0)}(1+|\nabla u|^2)^{\frac{q}{2}}\;dz\right]^{\frac 1q\frac{2q}{4-\hat N(q-p)}}, \end{aligned}NEWLINE\]NEWLINENEWLINENEWLINE for some constant \(C=C(\hat N,p,q,\nu,L)\), where \(\hat N=N\) if \(N\geq 3\) and \(\hat N\) is any number in \((2,\frac{4}{q-p})\) if \(N=2\). NEWLINENEWLINENEWLINEIn the second part of the article, the authors investigate the Cauchy-Dirichlet problem associated with the parabolic equation (1) and given by NEWLINE\[NEWLINE \partial_t u-\mathrm{div}\, a(x,t,\nabla u)=0\text{ in }\Omega_T,\;\; u=g\text{ on }\partial_P\Omega_T, \tag{4}NEWLINE\]NEWLINE where \(\partial_P\Omega_T=(\partial\Omega\times (0,T))\cup (\Omega\times\{0\})\) is the parabolic boundary of \(\Omega_T\). A function \(u\in L^p((0,T); W^{1,p}(\Omega))\cap L_{\mathrm{loc}}^q((0,T); W_{\mathrm{loc}}^{1,q}(\Omega))\) is said to be a weak solution of Problem (4) if (3) holds, \(u\in g+L^p((0,T);W_0^{1,p}(\Omega))\) and \(u(\cdot,0)=g(\cdot,0)\) in the \(L^2\)-sense. Some existence results and some gradient estimates of weak solutions to (4) have been obtained.
scientific article; zbMATH DE number 6244277
scientific article