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4 votes
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Continuity of weak solutions to wave equation with time-dependent coefficients

Consider the following second-order wave equation $$ u_{tt} - div( a\cdot \nabla u) = f \quad \text{ in } (0,T)\times \Omega $$ with boundary conditions $$ u(0)=g, \ u_t(0)=h, \ u|_{\partial \Omega}=0....
daw's user avatar
  • 273
4 votes
0 answers
613 views

well-posedness of the transport equation

I asked this question before on math exchange but did not have any luck with an answer. I would like to consider a simple example but get a thorough understanding of the theory behind it. I am ...
Kamil's user avatar
  • 153
3 votes
0 answers
208 views

Analytic solution to two component, first order, linear PDE system

I would like to obtain analytic solutions to the following PDE system: \begin{equation} \rho_t + D(\lambda)\,\rho_\lambda = A(\lambda) \rho, \tag{1} \end{equation} with $\rho = (\rho_0,\rho_1)^T$, $D$ ...
Frits Veerman's user avatar
3 votes
0 answers
338 views

Method of characteristic for a system of first order PDEs

I am working with this system of first order PDEs: \begin{equation} \left\{ \begin{aligned} %Suscettibili &\frac{\partial{S}(a,t)}{\partial{t}} + \frac{\partial{S}(a,t)}{\partial{a}}= -\lambda(a,...
CrishaD's user avatar
  • 31
2 votes
0 answers
60 views

Decay of solution for linear system with damping

Let us consider the following linear system with damping: $$ \begin{cases} u_t - u_x = -\frac{1}{2} (u+v)\\ v_t + v_x = -\frac{1}{2} (u+v) \end{cases} $$ Let's write the solution as $w=(u,v)$ ...
Riku's user avatar
  • 839
2 votes
0 answers
58 views

Convex solutions of linear hyperbolic PDEs in a planar domain

Consider a linear homogeneous 2nd-order PDE in a convex planar domain $\Omega$ : $$a(x,y)\frac{\partial^2u}{\partial x^2}+2b(x,y)\frac{\partial^2u}{\partial x\partial y}+c(x,y)\frac{\partial^2u}{\...
Denis Serre's user avatar
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2 votes
0 answers
50 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 ...
Gustave's user avatar
  • 617
1 vote
0 answers
39 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 ...
Gustave's user avatar
  • 617
1 vote
0 answers
106 views

Solution to hyperbolic linear second order PDE

I am trying to prove the existence (and uniqueness) of a weak solution for a specific PDE. First, let me formulate the problem. I asked the question on the Mathematics page but did not get a solution ...
SebastianP's user avatar
1 vote
0 answers
70 views

Smoothing in linear hyperbolic equations

This is a bit fuzzy, but I've somewhere read or heard something like: "For linear hyperbolic equations smoothing in time leads to smoothing in space" Is this in any sense true? References, ...
F.M.R.'s user avatar
  • 43
0 votes
0 answers
69 views

Inside and up to boundary regularity improvement of linear differential operator

I'm learning elliptic PDEs and a natural question came to me. Consider a constant coefficient linear differential operator defined on the ball $B_r:=\{\sum_{k=1}^n|x_k|^2<r\}$ $$A=\sum a_\alpha\...
Holden Lyu's user avatar