Mathematica 12.0 does the job by

    Integrate[Exp[\[Lambda]*t^4],{t, \[Alpha], 0},  Assumptions->\[Alpha]<0 && \[Lambda] >= 1]*
    Integrate[Exp[-\[Lambda]*t^4],{t,-Infinity,\[Alpha]},Assumptions->\[Alpha]<0&&\[Lambda]>=1]
$$-\frac{(-1)^{3/4} \alpha  E_{\frac{3}{4}}\left(\alpha ^4 \lambda \right) \left(\Gamma \left(\frac{1}{4},-\alpha ^4 \lambda \right)-\Gamma \left(\frac{1}{4}\right)\right)}{16 \sqrt[4]{\lambda }}  $$


    NMaximize[{%,\[Alpha]<0&&\[Lambda]>=1},\[Alpha],\[Lambda]},Method-> "DifferentialEvolution"]
$$\{0.209323,\{\alpha \to -0.476784,\lambda \to 1.\}\}$$

Addition. Maple confirms it by

    DirectSearch:-Search((alpha, lambda) -> int(exp(-lambda*t^4), t = -infinity .. alpha, numeric)*int(exp(lambda*t^4), t = alpha .. 0, numeric), {-100 <= alpha, 1 <= lambda, alpha <= 0, lambda <= 100}, maximize);
$$[ 0.209323347704846, \left[ \begin {array}{c} - 0.476781454615864297
\\ 1.00000000002946488\end {array} \right] ,117]
 $$