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Results tagged with ap.analysis-of-pdes
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user 493556
Partial differential equations (PDEs): Existence and uniqueness, regularity, boundary conditions, linear and non-linear operators, stability, soliton theory, integrable PDEs, conservation laws, qualitative dynamics.
4
votes
1
answer
177
views
On a non-linear PDE $p_t = e^{-p}p_{xx}$
Consider the PDE for $p:[0,T]\times [0,1]\to\mathbb R$ as follows:
$$
\begin{cases}
p_t = e^{-p}p_{xx}, & (t,x)\in (0,T)\times (0,1),\\
p(0,\cdot)\equiv -c &\text{for }x\in (0,1)\\
p(\cdot,0)\equiv 0 …
- 1,409
3
votes
0
answers
190
views
A non-linear PDE $v^2v_t=v_{xx}v-v_{x}^2$
PS : Indeed, there is a typo in my equation. Thanks to Zachary's observation.
Consider a PDE for $v: [0,1]^2\to (-\infty,0]$ satisfying
$$v_t(t,x) = \frac{v_{xx}(t,x)v(t,x)-v_{x}(t,x)^2}{v(t,x)^2},\qu …
- 1,409
3
votes
0
answers
117
views
Wellposedness of this parabolic PDE
Consider a terminal-boundary value problem for $v: (t,x,y)\in [0,T]\times \mathbb R^2_+\to \mathbb R\ni v(t,x,y)$:
$$
\begin{cases}
v_t + \max(v_x,v_y)+ \frac 1 2 (v_{xx}+v_{yy})=0, & \forall (t,x,y) …
- 1,409
2
votes
0
answers
69
views
Reference request : A SPDE model
Let $\Omega_0\subset\mathbb R^d$ be open and bounded with sufficiently smooth boundary $\partial\Omega_0$. Let $O\subset \Omega_0$ be a random open subset. Set $\Omega:=\Omega_0\setminus O$. Consider …
- 1,409
1
vote
0
answers
43
views
Wellposdeness of some HJB equation
Consider the non-linear PDE for $u:[0,1]\times [-1,1]\to\mathbb R$ as follows:
$$u_t= \inf_{b\ge 1/e} \big(-b u_{xx} - \log b - 1\big), \quad \forall (t,x) \in (0,1) \times (-1,1),$$
together with the …
- 1,409