Lectures on Partial Differential Equations

July, 25 at 9:30 a.m.
In
Antonio Gilioli Hall: Building A - Instituto de Matemática e Estatística - USP.

9:30 - José M. Arrieta - U. Complutense de Madrid

Nonautonomous degenerate logistic equations

We study the behavior of positive solutions of a logistic equation of the type $u_t-\Delta u= \lambda u- n(x,t)u^\rho$ in a bounded smooth domain with some homogeneous boundary conditions. We assume $\rho>1$ and $n(x,t)\geq 0$.  We want to understand how the relation between the value of the parameter $\lambda$ and the geometry of the set $K_0(t)=\{x\in \Omega: n(x,t)=0\}$ affects the global behavior of the solutions of the equation.  If $n(x,t)$ is independent of $t$, then we know from previous works that there is a critical value $\lambda_0(K_0)>0$ such that if $\lambda<\lambda_0(K_0)$ the solutions are globally bounded, while if $\lambda>\lambda_0(K_0)$ the solutions are unbounded as $t\to +\infty$. In the nonautonomous case, we will also find regimes in which the solutions are globally bounded and others in which the solutions are unbounded as $t\to +\infty$.
This is a joint work with Marcos Molina (Univ Complutense de Madrid) and Lucas Araujo Santos (Univ Federal de Paraiba).

10:10 - Coffee

10:40 - Marcus Marrocos - UFABC

Eigenvalues of the Neumann Laplacian in SO(n)-invariant regions in

In this work we are concerned with the multiplicity of the eigenvalues of the Neumann Laplacian in regions of Rn+1 which are invariant under the natural action of SO(n). We give a positive answer (in the Neumann case) to a conjecture by V. Arnold on the transversality of the transformation given by the Dirichlet integral to the stratification in the space of quadratic forms according to the multiplicities of the eigenvalues. We show that, generically in the set of SO(n)-invariant, C2-regions, the action is irreducible in each eigenspace Ker(∆ + λ).

11:20 - Antoine Laurain - IME USP

Recent advances in nonsmooth shape optimization

We will see some recent results about distributed and boundary expressions of first and second order shape derivatives for several classes of nonsmooth domains such as Lipschitz domains or polygons. Depending on the type of nonsmoothness, different boundary
expressions can be derived from the distributed expressions, which requires a careful study of the regularity of the solution to the underlying PDE. We will show applications to shape Hessians for polygons and to level set methods.

12:00 - Lunch

14:00 - Alexandre N. de Carvalho - ICMC USP

On the Gradient Structure of a Non-autonomous Chafee–Infante Like Problem

In this work we prove that some non-autonomous scalar one dimensional semi-linear parabolic problems have an associated skew-product semigroup with gradient structure similar to that observed for the autonomous Chafee–Infante problem. The aim is to exhibit a non-autonomous problem for which the asymptotic dynamics can be fairly well described. The tools involved are symmetry, invariance, comparison and the lap-number.

14:40 - Alexandre Kawano - Poli USP

Uma aranha pode ouvir a posição da presa?

Consideramos o problema inverso de se localizar uma presa a partir de medições da vibração  tomadas em torno do centro de uma teia, onde uma aranha se localiza. A teia real, formada por um número finito de fios radiais e  circunferenciais, é modelada como uma membrana contínua, que é modelada por uma EDPL com coeficiente singular na origem. A carga transversal que descreve o impacto da presa é assumido na forma $g(t)f(x)$, em que $g$ é uma função conhecida do tempo e $f$ é o uma função desconhecida a ser identificada. Provamos um resultado de unicidade para a identificação de $f$ quando é conhecido o deslocamento transversal em um anel com raio exterior arbitrariamente pequeno centrado na origem da teia, durante um intervalo de tempo finito.

Trabalho realizado em conjunto com Antonino Morassi.

15:20 - Coffee

15:50 - Ma To Fu - ICMC USP

Dynamics of semilinear wave equations with locally distributed damping

In this talk we present some classical results on the existence of global attractors for 3D wave equations with critical forcing and locally distributed damping. Then we study the continuity of attractors with respect to an external force parameter.

16:30 - Sérgio Oliva - IME USP

TBA