arXiv subjects

Gael Diebou Yomgne

Publications and source records attributed to Gael Diebou Yomgne.

3 recordsLinked to original sources

arXiv2021

On the generalized parabolic Hardy-Hénon equation: Existence, blow-up, self-similarity and large-time asymptotic behaviour

This paper deals with the Cauchy problem for the Hardy-Hénon equation (and its fractional analogue). Local well-posedness for initial data in the class of continuous functions with slow decay at infinity is investigated. Small data (in critical weak-Lebesgue space) global well-posedness is obtained in $C_{b}([0,\infty);L^{q_c,\infty}(\RR^{n}))$. As a direct consequence, global existence for data in strong critical Lebesgue $L^{q_c}(\RR^{n})$ follows under a smallness condition while uniqueness is unconditional. Besides, we prove the existence of self-similar solutions and examine the long time behavior of globally defined solutions. The zero solution $u\equiv 0$ is shown to be asymptotically stable in $L^{q_c}(\RR^{n})$ -- it is the only self-similar solution which is initially small in $L^{q_c}(\RR^{n})$. Moreover, blow-up results are obtained under mild assumptions on the initial data and the corresponding Fujita critical exponent is found.

math.AP↗

arXiv2021

Dirichlet problem for weakly harmonic maps with rough data

Weakly harmonic maps from a domain $Ω$ (the upper half-space $\Rd$ or a bounded $C^{1,α}$ domain, $α\in (0,1]$) into a smooth closed manifold are studied. Prescribing small Dirichlet data in either of the classes $L^{\infty}(\partialΩ)$ or $BMO(\partialΩ)$, we establish solvability of the resulting boundary value problems by means of a nonvariational method. As a by-product, solutions are shown to be locally smooth, $C^{\infty}_{loc}$. Moreover, we show that boundary data can be chosen large in the underlying topologies if $Ω$ is smooth and bounded by perturbing strictly stable smooth harmonic maps.

math.AP↗

arXiv2021

On a nonlinear Laplace equation related to the boundary Yamabe problem in the upper-half space

We consider in this paper the nonlinear elliptic equation with Neumann boundary condition \begin{align*} \begin{cases} Δu=a|u|^{m-1}u\,\,\mbox{ in }\,\,\rnp\\ \dfrac{\partial u}{\partial t}=b|u|^{η-1}u+f\,\,\mbox{ on }\,\,\partial\rnp. \end{cases} \end{align*} For $a,b\neq 0$, $m>\frac{n+1}{n-1}$, $(n>1)$, $η=\frac{m+1}{2}$ and small data $f\in L^{\frac{nq}{n+1},\infty}(\partial\rnp)$, $q=\frac{(n+1)(m-1)}{m+1}$ we prove that the problem is solvable. More precisely, we establish existence, uniqueness and continuous dependence of solutions on the boundary data $f$ in the function space $\X^{q}_{\infty}$ where \[\|u\|_{\X^{q}_{\infty}}=\sup_{t>0}t^{\frac{n+1}{q}-1}\|u(\cdot,t)\|_{L^{\infty}(\partial\rnp)}+\|u\|_{L^{\frac{q(m+1)}{2},\infty}(\rnp)}+\|\nabla u\|_{L^{q,\infty}(\rnp)}. \] As a direct consequence, we obtain the local regularity property $C^{1,ν}_{loc}$, $ν\in (0,1)$ of these solutions as well as energy estimates for certain values of $m$. Boundary values decaying faster than $|x|^{-(m+1)/(m-1)}$, $x\in \rn\setminus\{0\}$ yield solvability and this decay property is shown to be sharp for positive nonlinearities. Moreover, we are able to show that solutions inherit qualitative features of the boundary data such as positivity, rotational symmetry with respect to the $(n+1)$-axis, radial monotonicity in the tangential variable and homogeneity. When $a,b>0$, the critical exponent $m_c$ for the existence of positive solutions is identified, $m_c=(n+1)/(n-1)$.

math.AP↗