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I want to solve the equation: $$ \begin{cases} \nabla \times (A \mathbf v)=f, \quad x\in \Omega \\ \operatorname{div}(\mathbf v)=0, \end{cases} $$ where $\Omega \subset\mathbb{R}^n$, is an open set, …

Consider the hyperbolic equation on a rectangular domain of the form $(0, L_x) \times (0, L_y)$: $$ a^2 u_{xx} - b^2 u_{yy} = f(x, y), $$ with Dirichlet boundary conditions on $u$. By using the separa …

I want to solve the equation: $$ \begin{cases} \nabla \times (F\times\mathbf v)=g, \\ \operatorname{div}(\mathbf v)=0, \end{cases}\label{1}\tag{1} $$ where $F$ and $g$ are given vector fields. The th …

I want to now whether this equation is of hyperbolic type: $$(1-\partial_{xx})y_{tt}+y_{xxxx}=0$$ with boundary conditions $$y(t,0)=y(t,1)=y_x(t,0)=y_x(t,1)=0.$$ I would say that the answer is yes. By …

I want to solve the following system $$\eqalign{ & y_t = -y_x + z\text{ in }(0,\text{T}) \times (0,1) \cr & z_t = z_x + y\text{ in }(0,\text{T}) \times (0,1) \cr & y(0,x) = y_0,\,\,z(0,x) = z_0, …

I want to ask about the observability inequality for the heat equation ( internal control), we consider the backward heat equation with Dirichlet boundary conditions: \begin{array}{c} \varphi _{t}+\ …

I want to apply the semigroup approach of nonautonomous evolution equation for the following wave equation $$u'' - \Delta u + \int\limits_0^t {g(s)} \Delta u(s)ds = 0$$ This problem can be written und …