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"]

$$\{0.205391,\{\alpha \to -0.457146,\lambda \to 1.\}\} $$

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);

$$[ 0.205391328549229, \left[ \begin {array}{c} - 0.456953669173581544 \\ 1.00000000031899860\end {array} \right] ,103] $$

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);

$$[ 0.209323347704846, \left[ \begin {array}{c} - 0.476781454615864297 \\ 1.00000000002946488\end {array} \right] ,117] $$

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"]

$$\{0.205391,\{\alpha \to -0.457146,\lambda \to 1.\}\} $$

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"]

$$\{0.205391,\{\alpha \to -0.457146,\lambda \to 1.\}\} $$

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);

$$[ 0.205391328549229, \left[ \begin {array}{c} - 0.456953669173581544 \\ 1.00000000031899860\end {array} \right] ,103] $$