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]
$$-\frac{1}{4} \alpha  E_{\frac{3}{4}}\left(\alpha ^4 \lambda \right) \left(\frac{1}{4} \alpha  E_{\frac{3}{4}}\left(\alpha ^4 \lambda \right)+\frac{\Gamma \left(\frac{5}{4}\right)}{\sqrt[4]{\lambda }}\right)  $$

    NMaximize[{%,\[Alpha]<0&&\[Lambda]>=1},\[Alpha],\[Lambda]},Method-> "DifferentialEvolution"]
$$\{0.205391,\{\alpha \to -0.457146,\lambda \to 1.\}\} $$