[Paper Review] Local well-posedness and global existence for the biharmonic heat equation with exponential nonlinearity

Mohamed Majdoub, Sarah Otsmane+1 more|arXiv (Cornell University)|Jun 23, 2016

Advanced Mathematical Physics Problems6 citations

TL;DR

This paper establishes local well-posedness and global existence for the biharmonic heat equation with exponential nonlinearity $ f(u) hicksim e^{u^2} $ in Orlicz spaces $ \exp L^2 $. Using fixed-point arguments in $ \exp L^2_0 $ and $ \exp L^2 $, it proves existence of unique local solutions and global solutions under small initial data and power-like growth near zero, with decay estimates derived via $ L^p-L^q $ estimates for the biharmonic semigroup.

ABSTRACT

In this paper we prove local well-posedness in Orlicz spaces for the biharmonic heat equation $\partial_{t} u+ Δ^2 u=f(u),\;t>0,\;x\in\R^N,$ with $f(u)\sim \mbox{e}^{u^2}$ for large $u.$ Under smallness condition on the initial data and for exponential nonlinearity $f$ such that $f(u)\sim u^m$ as $u o 0,$ $m$ integer and $N(m-1)/4\geq 2$, we show that the solution is global. Moreover, we obtain a decay estimates for large time for the nonlinear biharmonic heat equation as well as for the nonlinear heat equation. Our results extend to the nonlinear polyharmonic heat equation.

Motivation & Objective

• To establish local well-posedness in Orlicz spaces $ \exp L^2_0(\mathbb{R}^N) $ for the biharmonic heat equation with exponential nonlinearity $ f(u) \sim e^{u^2} $.
• To prove global existence of solutions under small initial data and power-type nonlinearity $ |f(u)| \sim |u|^m $ near zero, with $ m \geq 2 $ and $ N(m-1)/4 \geq 2 $.
• To derive large-time decay estimates for both the nonlinear biharmonic and standard heat equations using $ L^p-L^q $ estimates of the biharmonic semigroup.
• To extend the results to the general polyharmonic heat equation $ \partial_t u + (-\Delta)^d u = f(u) $ for $ d \geq 2 $.

Proposed method

• Use of the integral formulation $ u(t) = e^{-t\Delta^2}u_0 + \int_0^t e^{-(t-s)\Delta^2} f(u(s))\,ds $ to study the Cauchy problem.
• Application of fixed-point arguments in the space $ \exp L^2_0(\mathbb{R}^N) $ for local well-posedness and in $ L^\infty(0,\infty; \exp L^2) $ for global existence.
• Employment of $ L^p-L^q $ estimates for the biharmonic semigroup $ e^{-t\Delta^2} $, with bounds $ \|e^{-t\Delta^2}\varphi\|_q \leq \mathcal{H} t^{-\frac{N}{4}(\frac{1}{p}-\frac{1}{q})} \|\varphi\|_p $.
• Use of Orlicz space norms, particularly the Luxemburg norm $ \|u\|_{\exp L^2} $, to handle exponential growth of the nonlinearity.
• Application of Hölder and Young-type inequalities to control $ \|f(u)\|_r $ in terms of $ \|u\|_{\exp L^2} $, leveraging exponential integrability.
• Extension to polyharmonic operators via $ L^p-L^q $ estimates for $ e^{-t(-\Delta)^d} $, with scaling $ t^{-\frac{N}{2d}(\frac{1}{p}-\frac{1}{q})} $.

Experimental results

Research questions

• RQ1 Under what conditions does the biharmonic heat equation with exponential nonlinearity $ f(u) \sim e^{u^2} $ admit a local solution in Orlicz spaces?
• RQ2 What smallness condition on initial data ensures global existence for the biharmonic equation with power-type nonlinearity near zero?
• RQ3 What decay rates can be established for solutions of the nonlinear biharmonic heat equation as $ t \to \infty $?
• RQ4 Can the results for the biharmonic equation be extended to higher-order polyharmonic equations $ \partial_t u + (-\Delta)^d u = f(u) $?
• RQ5 How do the $ L^p-L^q $ estimates of the biharmonic semigroup influence the existence and decay of solutions?

Key findings

• Local well-posedness holds in $ \exp L^2_0(\mathbb{R}^N) $ for initial data $ u_0 \in \exp L^2_0(\mathbb{R}^N) $, with a unique solution $ u \in C([0,T]; \exp L^2_0) $.
• Global existence is established for small initial data $ \|u_0\|_{\exp L^2} \leq \varepsilon $, with $ \varepsilon > 0 $ sufficiently small, under the condition $ N(m-1)/4 \geq 2 $ and $ m \geq 2 $.
• Solutions satisfy decay estimates $ \|u(t)\|_p \leq C t^{-\sigma} $ with $ \sigma = \frac{1}{m-1} - \frac{N}{2dp} > 0 $, valid for $ p $ in a specified range depending on $ N $ and $ d $.
• The biharmonic semigroup $ e^{-t\Delta^2} $ is continuous at $ t=0 $ in $ \exp L^2_0(\mathbb{R}^N) $, but not in $ \exp L^2(\mathbb{R}^N) $.
• The results extend to the polyharmonic equation $ \partial_t u + (-\Delta)^d u = f(u) $, with $ d \geq 2 $, using analogous $ L^p-L^q $ estimates and fixed-point arguments.
• For $ f(u) \sim e^{u^2} $, the Orlicz space $ \exp L^2 $ is the natural framework, as it embeds in $ L^q $ for all $ q \geq 2 $, and is invariant under the semigroup action.