Related papers: Inverse problems for a quasilinear hyperbolic equa…

The classical result of A. Ambrosetti and G. Prodi [1], in the form of M.S. Berger and E. Podolak [4], gives the exact number of solutions for the problem \[ \Delta u+g(u)= \mu \phi _1(x)+e(x) \;\; \mbox{in $D$} , \;\; u=0 \;\; \mbox{on…

The purpose of this paper is to investigate the well-posedness of several linear and nonlinear equations with a parabolic forward-backward structure, and to highlight the similarities and differences between them. The epitomal linear…

The existence problem is solved, and global pointwise estimates of solutions are obtained for quasilinear and Hessian equations of Lane-Emden type, including the following two model problems: $ -\Delta_p u = u^q + \mu$ and $F_k[-u] = u^q +…

We prove the existence and uniqueness of solution of the obstacle problem for quasilinear stochastic partial differential equations (OSPDEs for short) with Neumann boundary condition. Our method is based on the analytical technics coming…

We develop computer-assisted tools to study semilinear equations of the form \begin{equation*} -\Delta u -\frac{x}{2}\cdot \nabla{u}= f(x,u,\nabla u) ,\quad x\in\mathbb{R}^d. \end{equation*} Such equations appear naturally in several…

We use the framework of Colombeau algebras of generalized functions to study existence and uniqueness of global generalized solutions to mixed non-local problems for a semilinear hyperbolic system. Coefficients of the system as well as…

This paper considers the weakly coupled parabolic system $\partial_t u-\partial^2_xu +P(x)u=0$ with the homogeneous Neumann boundary condition, where \(P(x)\) is a \(2\times2\) symmetric real-valued function matrix. Under the assumption…

We consider inverse problems in space-time $(M, g)$, a $4$-dimensional Lorentzian manifold. For semilinear wave equations $\square_g u + H(x, u) = f$, where $\square_g$ denotes the usual Laplace-Beltrami operator, we prove that the…

We prove the existence and uniqueness of strong solutions to the equation $u u_x - u_{yy} = f$ in the vicinity of the linear shear flow, subject to perturbations of the source term and lateral boundary conditions. Since the solutions we…

This paper investigates an inverse boundary value problem for a semilinear strongly damped wave equation with Dirichlet boundary conditions in Sobolev spaces of functions bounded in time on $\R$, including periodic and almost periodic…

Reaction-diffusion equations are one of the most common partial differential equations used to model physical phenomenon. They arise as the combination of two physical processes: a driving force $f(u)$ that depends on the state variable $u$…

In this paper, we are concerned with the quasilinear Schr\"{o}dinger equation \begin{equation*} -\Delta u+V(x)u-u\Delta(u^2)=g(u),\ \ x\in \mathbb{R}^{N}, \end{equation*} where $N\geq3$, $V$ is radially symmetric and nonnegative, and $g$ is…

In this work, we consider the inverse problem of simultaneously recovering two classes of quasilinear terms appearing in a parabolic equation from boundary measurements. It is motivated by several industrial and scientific applications,…

We present in this note a local in time well-posedness result for the singular $2$-dimensional quasilinear generalized parabolic Anderson model equation $$ \partial_t u - a(u)\Delta u = g(u)\xi $$ The key idea of our approach is a simple…

In this paper, we study the quasilinear inequality $ \Delta_m u+f(u)\leq 0$ on a complete Riemannian manifold, where \begin{align*} m>1,\alpha>m-1 \quad and \quad f(t)> 0,\alpha f(t)-tf^{'}(t)\geq 0, \forall t>0. \end{align*} If for some…

We consider the problem of existence of a solution $u$ to $\partial_t u-\partial_{xx} u = 0$ in $(0,T)\times\mathbb{R}_+$ subject to the boundary condition $-u_x(t,0)+g(u(t,0))=\mu$ on $(0,T)$ where $\mu$ is a measure on $(0,T)$ and $g$ a…

In this paper we establish uniqueness in the inverse boundary value problem for the two coefficients in the inhomogeneous porous medium equation $\epsilon\partial_tu-\nabla\cdot(\gamma\nabla u^m)=0$, with $m>1$, in dimension 3 or higher,…

Let $\Omega\subseteq \mathbb{R}^N$ a bounded open set, $N\geq 2$, and let $p>1$; we prove existence of a renormalized solution for parabolic problems whose model is $$ \begin{cases} u_{t}-\Delta_{p} u=\mu & \text{in}\…

Suppose $q_i(x)$, $i=1,2$ are smooth functions on $\R^3$ and $U_i(x,t)$ the solutions of the initial value problem {gather*} \pa_t^2 U_i- \Delta U_i - q_i(x) U_i = \delta(x,t), \qquad (x,t) \in \R^3 \times \R U_i(x,t) =0, \qquad \text{for}…

In this paper, we establish the existence of a solution for a class of quasilinear equations characterized by the prototype: \begin{equation} \left\{\begin{aligned} -\operatorname{div}(\vartheta_\alpha|\nabla u|^{p-2} \nabla…