Lecture 1: We discuss various methods to obtain sharp existence and classification results for the positive solutions of nonlinear elliptic equations in $\mathbb R^N\setminus \{0\}$. Problems of this type have been studied extensively by many authors. For the prototype model $\Delta u=u^q$ in $\mathbb R^N\setminus \{0\}$ with $N\geq 3$, the pioneering paper of Brezis–Véron (1980/1981) shows that there are no positive solutions when $q\geq N/(N-2)$. When $1<q<N/(N-2)$, then the existence and profile near zero of all positive $C^1(\mathbb R^N\setminus\{0\})$-solutions are given by Friedman and Véron (1986). We provide the counterpart of these results for elliptic equations featuring a Hardy potential and weighted nonlinearities. Lecture 1 is based on joint work with M. Fărcăşeanu [J. Differential Equations 292 (2021), 461–500].

Lecture 2: We consider the positive solutions to the model problem $\Delta u=u^q |\nabla u|^m$ in $\Omega\setminus \{0\}$, where $\Omega$ is a domain in $\mathbb R^N$ $(N\geq 2)$ with $0\in \Omega$. We assume that $q\geq 0$, $m\in (0,2)$ and $m+q>1$. We present methods to obtain a complete classification of the behaviour near zero, as well as at $\infty$ when $\Omega=\mathbb R^N$, of the positive solutions of the above problem, together with corresponding existence results. This study is motivated by a rich literature on the topic of isolated singularities (e,g, Serrin (1965), Brezis–Oswald (1987), Vázquez–Véron (1980; 1985); Véron (1981; 1986; 1996), Nguyen Phuoc–Véron (2012) and Marcus–Nguyen (2015)). We emphasise the changes that arise from the introduction of the gradient factor in the nonlinear term and the new phenomena emerging when $m\in (0,1)$. Lecture 2 is based on joint work with J. Ching [Anal. PDE, 8 (8) (2015), 1931–1962].

We consider boundary value problems of the form $-(\Delta + V )u + f(u) = \tau$ in $D\subset \mathbb R^N$, $\mathrm{tr}_V u = \nu$ on $\partial D$ where $D$ is a bounded Lipschitz domain in $\mathbb R^N$ and $\mathrm{tr}_V u$ denotes the measure boundary trace associated with $V$. Regarding the non-linear term assume: $f$ is continuous, monotone increasing and $f(0) = 0$. We discuss questions of existence and uniqueness, first in the case $V = 0$ and then for potentials $V$ that blow up at the boundary not faster then $\mathrm{dist}(x; \partial D)^{-2}$.

In this talk, I will present a Schauder-type estimate for general local and non-local linear parabolic system $$\partial_tu+\mathcal{L}_su=\Lambda^\gamma f+g$$ in $(0,\infty)\times\mathbb{R}^d$ where $\Lambda=(-\Delta)^{\frac{1}{2}}$, $0<\gamma\leq s$, $\mathcal{L}_s$ is the Pesudo-differential operator of the order $s$. By applying our Schauder-type estimate to suitably chosen differential operators $\mathcal{L}_s$, we obtain critical well-posedness results of various local and non-local nonlinear evolution equations in geometry and fluids, including hypoviscous Navier–Stokes equations, the surface quasi- geostrophic equation, mean curvature equations, Willmore flow, surface diffusion flow, Peskin equations, thin-film equations and Muskat equations.

I will present some problems related to the occurrence/appearance of singularities in viscous Hamilton-Jacobi equations with coercive superlinear first order terms, in both stationary and time-dependent equations. The main focus will be on boundary blow-up solutions and barrier type phenomena, which have a substantial role in the formation of singularities of time-dependent classical solutions.

We give an overview of the old and more recent developments of the study of the singularity problem for quasilinear elliptic equations in a domain of $\mathbb{R}^N$ $$ -div A(x,u,\nabla u)+B(x,u,\nabla u)=0 $$ since the pioneering works of James Serrin (1964-1965). The problem is twofold:
1- If the above equation is satisfied in a punctured domain say $B_1\setminus\{0\}$, is it possible to describe the behaviour of $u(x)$ when $x\to 0$ ?
2- If the above equation is satisfied in $B_1\setminus \Sigma$ where $\Sigma$ is a subset of $B_1$, under what conditions the function can be extended as a solution of the same equation in whole $\Omega$ (we say that $\Sigma$ is a removable singularity) ?
Examples are $$A(x,u,\nabla u)=|\nabla u|^{p-2}\nabla u$$ and $$B(x,u,\nabla u)=\pm |u|^{q-1}u\;±, \;B(x,u,\nabla u)= \pm |\nabla u|^r\quad \text{or }\;B(x,u,\nabla u)= |u|^{q-1}u\pm |\nabla u|^r.$$ We will recall that Serrin's assumptions are (with $1<m\leq N$) $$A(x,u,\nabla u)\sim |\nabla u|^{m-2}\nabla u\, \text{ and }\;|B(x,u,\nabla u)|\leq c(|u|^{m-1}+ |\nabla u|^{m-1}), $$ and in his case the pertubation term $B$ plays a minor role. This is the contrary in the two fundamental superlinear cases that we will present: Lane-Emden's equation $ -\Delta u-u^q=0 $ and Emden-Fowler's equations $ -\Delta u+u^q=0 $ where $q>1$ and $u\geq 0$.

I will give lectures on recent advances in parabolic gluing methods and applications to Type II blowup for energy-critical Fujita equation.

The dynamical system approach has been well applied to study the traveling wave solutions of various nonlinear wave equations in recent two decades. In this talk, the main ideal and the key steps of this approach to investigate the traveling wave solutions of nonlinear wave equations will be introduced firstly. In addition to the exact solutions, the bifurcations of the traveling wave solutions and various smooth or non-smooth can be derived via this approach. The advantage and the challenging of this method is discussed as well. How this approach is extended to examine the traveling waves solutions, even the multi-wave solutions of higher-order nonlinear wave equations will be presented. Some recent development and works on the persistence of traveling wave solutions and the new solitary wave solutions to some classical nonlinear wave equations under singular perturbation via combining the singular perturbation method and bifurcation analysis are presented.

Lecture 1: We discuss various methods to obtain sharp existence and classification results for the positive solutions of nonlinear elliptic equations in $\mathbb R^N\setminus \{0\}$. Problems of this type have been studied extensively by many authors. For the prototype model $\Delta u=u^q$ in $\mathbb R^N\setminus \{0\}$ with $N\geq 3$, the pioneering paper of Brezis–Véron (1980/1981) shows that there are no positive solutions when $q\geq N/(N-2)$. When $1<q<N/(N-2)$, then the existence and profile near zero of all positive $C^1(\mathbb R^N\setminus\{0\})$-solutions are given by Friedman and Véron (1986). We provide the counterpart of these results for elliptic equations featuring a Hardy potential and weighted nonlinearities. Lecture 1 is based on joint work with M. Fărcăşeanu [J. Differential Equations 292 (2021), 461–500].

Lecture 2: We consider the positive solutions to the model problem $\Delta u=u^q |\nabla u|^m$ in $\Omega\setminus \{0\}$, where $\Omega$ is a domain in $\mathbb R^N$ $(N\geq 2)$ with $0\in \Omega$. We assume that $q\geq 0$, $m\in (0,2)$ and $m+q>1$. We present methods to obtain a complete classification of the behaviour near zero, as well as at $\infty$ when $\Omega=\mathbb R^N$, of the positive solutions of the above problem, together with corresponding existence results. This study is motivated by a rich literature on the topic of isolated singularities (e,g, Serrin (1965), Brezis–Oswald (1987), Vázquez–Véron (1980; 1985); Véron (1981; 1986; 1996), Nguyen Phuoc–Véron (2012) and Marcus–Nguyen (2015)). We emphasise the changes that arise from the introduction of the gradient factor in the nonlinear term and the new phenomena emerging when $m\in (0,1)$. Lecture 2 is based on joint work with J. Ching [Anal. PDE, 8 (8) (2015), 1931–1962].

We consider boundary value problems of the form $-(\Delta + V )u + f(u) = \tau$ in $D\subset \mathbb R^N$, $\mathrm{tr}_V u = \nu$ on $\partial D$ where $D$ is a bounded Lipschitz domain in $\mathbb R^N$ and $\mathrm{tr}_V u$ denotes the measure boundary trace associated with $V$. Regarding the non-linear term assume: $f$ is continuous, monotone increasing and $f(0) = 0$. We discuss questions of existence and uniqueness, first in the case $V = 0$ and then for potentials $V$ that blow up at the boundary not faster then $\mathrm{dist}(x; \partial D)^{-2}$.

In this talk, I will present a Schauder-type estimate for general local and non-local linear parabolic system $$\partial_tu+\mathcal{L}_su=\Lambda^\gamma f+g$$ in $(0,\infty)\times\mathbb{R}^d$ where $\Lambda=(-\Delta)^{\frac{1}{2}}$, $0<\gamma\leq s$, $\mathcal{L}_s$ is the Pesudo-differential operator of the order $s$. By applying our Schauder-type estimate to suitably chosen differential operators $\mathcal{L}_s$, we obtain critical well-posedness results of various local and non-local nonlinear evolution equations in geometry and fluids, including hypoviscous Navier–Stokes equations, the surface quasi- geostrophic equation, mean curvature equations, Willmore flow, surface diffusion flow, Peskin equations, thin-film equations and Muskat equations.

I will present some problems related to the occurrence/appearance of singularities in viscous Hamilton-Jacobi equations with coercive superlinear first order terms, in both stationary and time-dependent equations. The main focus will be on boundary blow-up solutions and barrier type phenomena, which have a substantial role in the formation of singularities of time-dependent classical solutions.

I will give lectures on recent advances in parabolic gluing methods and applications to Type II blowup for energy-critical Fujita equation.

The dynamical system approach has been well applied to study the traveling wave solutions of various nonlinear wave equations in recent two decades. In this talk, the main ideal and the key steps of this approach to investigate the traveling wave solutions of nonlinear wave equations will be introduced firstly. In addition to the exact solutions, the bifurcations of the traveling wave solutions and various smooth or non-smooth can be derived via this approach. The advantage and the challenging of this method is discussed as well. How this approach is extended to examine the traveling waves solutions, even the multi-wave solutions of higher-order nonlinear wave equations will be presented. Some recent development and works on the persistence of traveling wave solutions and the new solitary wave solutions to some classical nonlinear wave equations under singular perturbation via combining the singular perturbation method and bifurcation analysis are presented.

Lecture 1: We discuss various methods to obtain sharp existence and classification results for the positive solutions of nonlinear elliptic equations in $\mathbb R^N\setminus \{0\}$. Problems of this type have been studied extensively by many authors. For the prototype model $\Delta u=u^q$ in $\mathbb R^N\setminus \{0\}$ with $N\geq 3$, the pioneering paper of Brezis–Véron (1980/1981) shows that there are no positive solutions when $q\geq N/(N-2)$. When $1<q<N/(N-2)$, then the existence and profile near zero of all positive $C^1(\mathbb R^N\setminus\{0\})$-solutions are given by Friedman and Véron (1986). We provide the counterpart of these results for elliptic equations featuring a Hardy potential and weighted nonlinearities. Lecture 1 is based on joint work with M. Fărcăşeanu [J. Differential Equations 292 (2021), 461–500].

Lecture 2: We consider the positive solutions to the model problem $\Delta u=u^q |\nabla u|^m$ in $\Omega\setminus \{0\}$, where $\Omega$ is a domain in $\mathbb R^N$ $(N\geq 2)$ with $0\in \Omega$. We assume that $q\geq 0$, $m\in (0,2)$ and $m+q>1$. We present methods to obtain a complete classification of the behaviour near zero, as well as at $\infty$ when $\Omega=\mathbb R^N$, of the positive solutions of the above problem, together with corresponding existence results. This study is motivated by a rich literature on the topic of isolated singularities (e,g, Serrin (1965), Brezis–Oswald (1987), Vázquez–Véron (1980; 1985); Véron (1981; 1986; 1996), Nguyen Phuoc–Véron (2012) and Marcus–Nguyen (2015)). We emphasise the changes that arise from the introduction of the gradient factor in the nonlinear term and the new phenomena emerging when $m\in (0,1)$. Lecture 2 is based on joint work with J. Ching [Anal. PDE, 8 (8) (2015), 1931–1962].

We consider boundary value problems of the form $-(\Delta + V )u + f(u) = \tau$ in $D\subset \mathbb R^N$, $\mathrm{tr}_V u = \nu$ on $\partial D$ where $D$ is a bounded Lipschitz domain in $\mathbb R^N$ and $\mathrm{tr}_V u$ denotes the measure boundary trace associated with $V$. Regarding the non-linear term assume: $f$ is continuous, monotone increasing and $f(0) = 0$. We discuss questions of existence and uniqueness, first in the case $V = 0$ and then for potentials $V$ that blow up at the boundary not faster then $\mathrm{dist}(x; \partial D)^{-2}$.

In this talk, I will present a Schauder-type estimate for general local and non-local linear parabolic system $$\partial_tu+\mathcal{L}_su=\Lambda^\gamma f+g$$ in $(0,\infty)\times\mathbb{R}^d$ where $\Lambda=(-\Delta)^{\frac{1}{2}}$, $0<\gamma\leq s$, $\mathcal{L}_s$ is the Pesudo-differential operator of the order $s$. By applying our Schauder-type estimate to suitably chosen differential operators $\mathcal{L}_s$, we obtain critical well-posedness results of various local and non-local nonlinear evolution equations in geometry and fluids, including hypoviscous Navier–Stokes equations, the surface quasi- geostrophic equation, mean curvature equations, Willmore flow, surface diffusion flow, Peskin equations, thin-film equations and Muskat equations.

I will present some problems related to the occurrence/appearance of singularities in viscous Hamilton-Jacobi equations with coercive superlinear first order terms, in both stationary and time-dependent equations. The main focus will be on boundary blow-up solutions and barrier type phenomena, which have a substantial role in the formation of singularities of time-dependent classical solutions.

I will give lectures on recent advances in parabolic gluing methods and applications to Type II blowup for energy-critical Fujita equation.

The dynamical system approach has been well applied to study the traveling wave solutions of various nonlinear wave equations in recent two decades. In this talk, the main ideal and the key steps of this approach to investigate the traveling wave solutions of nonlinear wave equations will be introduced firstly. In addition to the exact solutions, the bifurcations of the traveling wave solutions and various smooth or non-smooth can be derived via this approach. The advantage and the challenging of this method is discussed as well. How this approach is extended to examine the traveling waves solutions, even the multi-wave solutions of higher-order nonlinear wave equations will be presented. Some recent development and works on the persistence of traveling wave solutions and the new solitary wave solutions to some classical nonlinear wave equations under singular perturbation via combining the singular perturbation method and bifurcation analysis are presented.