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I've come across the following simple family of PDE's and am wondering if they fit into a better known class or if they are attack-able by any standard techniques. The equation for $u(x, t)$ with $t \geq 0,x \geq 1$ is $$ u_{tx} = - (1 - u)^n $$ with boundary conditions $$ \begin{cases} u_x(x, 0) = 1 - x \\ u(x, 0) = 0 \\ u(1, t) = 0 \end{cases} $$$$ \begin{cases} u_t(x, 0) = 1 - x \\ u(x, 0) = 0 \\ u(1, t) = 0 \end{cases} $$ with $n \geq 1$ an integer.

I've come across the following simple family of PDE's and am wondering if they fit into a better known class or if they are attack-able by any standard techniques. The equation for $u(x, t)$ with $t \geq 0,x \geq 1$ is $$ u_{tx} = - (1 - u)^n $$ with boundary conditions $$ \begin{cases} u_x(x, 0) = 1 - x \\ u(x, 0) = 0 \\ u(1, t) = 0 \end{cases} $$ with $n \geq 1$ an integer.

I've come across the following simple family of PDE's and am wondering if they fit into a better known class or if they are attack-able by any standard techniques. The equation for $u(x, t)$ with $t \geq 0,x \geq 1$ is $$ u_{tx} = - (1 - u)^n $$ with boundary conditions $$ \begin{cases} u_t(x, 0) = 1 - x \\ u(x, 0) = 0 \\ u(1, t) = 0 \end{cases} $$ with $n \geq 1$ an integer.

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A non-linear PDE

I've come across the following simple family of PDE's and am wondering if they fit into a better known class or if they are attack-able by any standard techniques. The equation for $u(x, t)$ with $t \geq 0,x \geq 1$ is $$ u_{tx} = - (1 - u)^n $$ with boundary conditions $$ \begin{cases} u_x(x, 0) = 1 - x \\ u(x, 0) = 0 \\ u(1, t) = 0 \end{cases} $$ with $n \geq 1$ an integer.