$L_1= \left\{ wxwy \;|\; x,w,y \in (a+b)^+ \right \}$
here $x,y \in(a+b)^+$ means any string having length $\geq 1$ ,( and not a big problem).
$w$ must be same at both place having length $\geq 1$ , start with $a$ or $b$
$ \overbrace{a(something)}^{w}\;x\;\overbrace{a\underbrace{(something)}_{ must\; be \; same \;as \; earlier}}^w\;y + \overbrace{b(something)}^{w}\;x\;\overbrace{b\underbrace{(something)}_{ must\; be \; same \;as \; earlier}}^w\;y$
Remember $x$ and $y$ can be anything having length $\geq 1$ , i.e, belong to $(a+b)^+$, we are flexible with the length of $x$ and $y$ , except $0$ length .
$ a\;\overbrace{(something)x}^{\in (a+b)^+}\;a\; \overbrace{(something)y}^{\in (a+b)^+} + \;b\;\overbrace{(something)x}^{\in (a+b)^+}\;b\; \overbrace{(something)y}^{\in (a+b)^+}$
here if something is not same, we can say that something is part of $x$ and $y$ respectively and $w$ is simple $a$ or $b$, rest is $x$ and $y$.
$ a(a+b)^+a(a+b)^+ + b(a+b)^+b(a+b)^+$
In simple words , here $x$ and $y$ can absorb rest of the strings
$L_1$ is Regular.
$L_2= \left\{ xwyw \;|\; x,w,y \in (a+b)^+ \right \}$
$w$ must be same at both place having length $\geq 1$ , end with $a$ or $b$
$ x\;\overbrace{(something)a}^{w}\;y\;\overbrace{\underbrace{(something)}_{ must\; be \; same \;as \; earlier}a}^w + \;x\;\overbrace{(something)b}^{w}\;y\;\overbrace{\underbrace{(something)}_{ must\; be \; same \;as \; earlier}b}^w$
$ \overbrace{x(something)}^{\in (a+b)^+}\;a\; \overbrace{y(something)}^{\in (a+b)^+}\;a + \overbrace{x(something)}^{\in (a+b)^+}\;b\; \overbrace{y(something)}^{\in (a+b)^+}\;b$
here if something is not same, we can say that something is part of $x$ and $y$ respectively and $w$ is simple $a$ or $b$, rest is $x$ and $y$.
$ (a+b)^+a(a+b)^+a + (a+b)^+b(a+b)^+b $
In simple words , here $x$ and $y$ can absorb rest of the strings
$L_2$ is Regular.
$L_3= \left\{ wxyw \;|\; x,w,y \in (a+b)^+ \right \}$
$w$ must be same at both place having length $\geq 1$ , start with $a$ or $b$
$ \overbrace{a(something)}^{w}\;x\;y\;\overbrace{a\underbrace{(something)}_{ must\; be \; same \;as \; earlier}}^w + \overbrace{b(something)}^{w}\;x\;y\;\overbrace{b\underbrace{(something)}_{ must\; be \; same \;as \; earlier}}^w$
$ a\;\overbrace{(something)x}^{\in(a+b)^+}\;y\;a\;\overbrace{\underbrace{(something)}_{ must\; be \; same \;as \; earlier}}^{ Problem} +b\;\overbrace{(something)x}^{\in(a+b)^+}\;y\;b\overbrace{\underbrace{(something)}_{ must\; be \; same \;as \; earlier}}^{ Problem}$
Problem is that, there is $x$ in right of first something , if both did not same we can say ,that can be part of $x$, but there is no $x$ or $y$ in left/right of second something, to absorb that. ( and there is $a$ or $b$ , that must be remain unchanged.)
$L_3$ is not regular.
