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Results tagged with hyperbolic-pde
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user 106804
Questions about partial differential equations of hyperbolic type. Often used in combination with the top-level tag ap.analysis-of-pdes.
1
vote
1
answer
220
views
Exact solution of two coupled transport equations
I want to solve the following system
$$\eqalign{
& y_t = -y_x + z\text{ in }(0,\text{T}) \times (0,1) \cr
& z_t = z_x + y\text{ in }(0,\text{T}) \times (0,1) \cr
& y(0,x) = y_0,\,\,z(0,x) = z_0, …
- 617
2
votes
0
answers
48
views
Proving the existence of solutions of a coupled wave equation with dynamical boundary condit...
I want to prove the existence of the solution of this system by using the Faedo-Galerkin approximation method, I have to choose a basis for working on and I don't know how to do it in this case, I sug …
- 617
3
votes
1
answer
265
views
Is this equation of hyperbolic type?
I want to now whether this equation is of hyperbolic type:
$$(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.$$
I would say that the answer is yes. By …
- 617
2
votes
0
answers
49
views
Nonautonomous wave equation of memory type
I want to apply the semigroup approach of nonautonomous evolution equation for the following wave equation
$$u'' - \Delta u + \int\limits_0^t {g(s)} \Delta u(s)ds = 0$$
This problem can be written und …
- 617
1
vote
0
answers
43
views
Hidden regularity for the coupled wave equation with dynamaic boundary condition
We have the equation
\begin{equation}
\left\{
\begin{array}{rrrr}
u_{tt}-\Delta u=0,&\text{in} &
\Omega \times ]0,T[ & \left( 1.1\right) \\
u=0, & \text{on
} & \Gamma _{0}\times ]0,T[ & \left( 1 …
- 617
2
votes
0
answers
61
views
Strong stability of the wave equation with time depending potential
It is well known that the wave equation with frictional damping
$$\eqalign{
& {y_{tt}} = {y_{xx}} - a(t,x){y_t}{\text{ }}{\text{,(t}}{\text{,x)}} \in {\text{ }}(0,\infty ) \times (0,1) \cr
& y …
- 617
1
vote
0
answers
28
views
Hyperbolic equation without initial state
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 …
- 617
1
vote
1
answer
239
views
Infinitesimal generator of a semigroup with parameter
When we talk about evolution equation the first idea that comes to the mind is the semigroups theory. This theory deals with Cauchy problems of the form $$\frac{{\partial u}}{{\partial t}} = Au,$$
whe …
- 617
2
votes
1
answer
196
views
Well-posedness of wave equations with time-dependent coefficient
Let us consider the following wave equation
\begin{array}{rrr}
y_{tt}=y_{xx}+a(t,x)y_{t} & \text{in}
& (0,T)\times (0,1), \\
y(0,t)=0\text{ , }y(t,1)=0 & \text{in} & (0,T), \\
y(0,x)=y_{0}(x)\text{ …
- 617
1
vote
0
answers
25
views
Stabilization of non-autonomuous 1-d wavs equation
I want to ask two questions about the stabilization of the equation $$\eqalign{
& {y_{tt}} = k(t,x){y_{xx}}+a(t,x){y_t}+ b(t,x){y_x}+ c(t,x){y_x} +d(t,x)y \ \ (t,x) \in {\text{ }}(0,\infty ) \ti …
- 617
1
vote
1
answer
172
views
Observability inequality for the 1D transport equation
Let $(a,b) \subset (0,1)$. Consider the following transport equation
$$z_t+z_x=0, \ (t,x)\in(0,T)\times(0,1), \\z(t,0)=0, \ z(0,x)=z_0(x).$$
It is clear that the solution to the above equation is give …
- 617