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Session 23. Nonlinear Evolution Equations and their Applications |
Regularity results for solutions to heat equation with the initial condition in Orlicz-Slobodetskii space |
| Agnieszka Kałamajska, University of Warsaw, Poland |
| The talk is based on joint works with Miroslav Krbec |
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We study the initial problem for heat equation:
\begin{equation}\label{heatt}
\begin{cases}
{u}_t(x,t) = \Delta_x{u}(x,t) &\text{in }\Omega\times (0,T),\\
{u}(x,0) =u_0 &\text{for } x\in \Omega,
\end{cases}
\end{equation}
where \(\Omega\subseteq \mathbb{R}^{n}\) is a Lipschitz boundary
domain, \(u\) lies in the completion of \(C_0^\infty (\Omega)\) in
certain Orlicz-Slobodetski type space \(Y^{{R_1},{R_2}}(\Omega)\)
which is defined in the following way. Let \(R_1,R_2\) be the
possibly different Orlicz spaces. By \(Y^{{R_1},{R_2}}(\Omega)\)
and denote the space consisting of all \(u\in L^{R_1}(\Omega)\), for
which the seminorm
\begin{equation}\label{defa}
I^{{R_2}} (u,\Omega):= \int_{\Omega}\int_{\Omega}{R_2} \left( \frac{| u(x) - u(y)| }{|x-y| }\right)\frac{dx dy}{|x-y|^{n-1}}
\end{equation}
is finite.
We prove that if \({R}\) satisfies certain assumptions and \(u_0\in Y^{R,R}(\Omega)\), then the solution \({u}\) of our heat equation lies in the Orlicz-Sobolev space \(W^{1,{R}}(\Omega\times (0,T))\), which by definition fulfills the requirement: \({u}\), together with its all first order partial derivatives belongs to Orlicz space \(L^{R} (\Omega\times (0,T))\). The typical representant of the admissible Orlicz space is \(L(LogL)^\alpha\) where \(\alpha\ge 1\). More generally, when \(R_1,R_2\) are possibly different but satisfy certain compatibility condition due to Kita, we obtain regularity results involving the initial condition \(u_0\in Y^{{R_1},{R_2}}(\Omega)\) and \(u\in W^{1,R_1}(\Omega\times (0,T))\), where \(R_1\) can essentially dominate \(R_2\). Lecture will be based on the following issues: |
References
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